Computational control for hybrid computation of hamiltonian eigen-solution

Abstract

Computational control for hybrid computation of Hamiltonian eigenstates can be provided by selecting K a number of basis states from a hypothetical space of a chemical system, wherein the hypothetical space is generated by a quantum computer system and includes fewer basis states than a complete basis space of the chemical system, wherein the number of basis states is selected according to a selection protocol to define a core space of the chemical system; computing eigenstates of the chemical system from the core space via an eigenstate solver provided by a classical computer system; and outputting the eigenstates of the chemical system. K ​

Description

[0001] Cross-reference to any priority claim Any and all applications that make priority claims in the PCT request are hereby incorporated in their entirety. Technical Field

[0002] This disclosure relates to computation using a quantum computer. Background Technology

[0003] quantum computer Unlike classical computers, quantum computers utilize quantum mechanical phenomena to store data in qubits (or "qubits"), which can store classical representations of "0," "1," or a superposition of both. Therefore, when a qubit is read, it doesn't output true / false or 1 / 0 like a classical bit; instead, it provides these classical values ​​as output based on the qubit's state, according to probability. Quantum computers have several promising applications thanks to the nondeterministic nature of quantum phenomena, offering greater speed and efficiency than classical computers in solving certain classes of computations. However, these quantum computers are susceptible to noise during computation, which can hinder them from deriving solutions to various computations (e.g., getting trapped in local minima / maxima, noise dominating the output value, etc.).

[0004] Calculate Hamiltonian One promising area for quantum computers is simulating the quantum properties of chemical systems, such as by calculating characteristic solutions to the system's Hamiltonian. The Hamiltonian corresponds to the total energy in the simulated system (e.g., the kinetic and potential energies of the constituent particles) and provides possible energies based on a set of eigenvalues ​​and eigenvectors. These values ​​have proven invaluable for researchers in assessing the potential chemical interactions between complex molecules and with biological structures (e.g., cells, viruses, and their components), and can be used to identify new drugs and therapeutics and their uses.

[0005] Do not acknowledge existing technology The discussion in this section is intended to provide background information for this disclosure and does not constitute an admission of prior art. Summary of the Invention

[0006] Hybrid computing This disclosure provides computational control for hybrid computation of Hamiltonian characteristic solutions. In a hybrid computing environment, both classical and quantum computing devices are used to provide the operator with the advantages of each computing technique while mitigating its disadvantages. For example, by controlling the computation of Hamiltonian characteristic solutions in a hybrid environment as described in this disclosure, the operator can produce a more accurate set of answers compared to using only a quantum computing system, and can arrive at the answer faster (and with fewer computational resources) compared to using only a classical computing system. Additionally, the additional functionality provided in this disclosure includes improvements to conventional hybrid computing systems by identifying when a given operation is performed in either a classical or quantum computing system.

[0007] handover algorithm Traditional quantum computing practices leverage the increased efficiency of quantum computers to potentially solve a variety of problems in chemical analysis that have proven difficult for classical computers to handle. Classical computers are traditionally prepared to run a problem on a quantum computer, and then the output of the quantum computation is decoded to obtain the answer. However, quantum computing remains a developing technology with limited hardware access, and the output provided by a quantum computer may not be accurate enough due to inherent noise. As described in this disclosure, the handover algorithm allows a classical computer to take over the computation from the quantum computer before it outputs a solution in the conventional way, but after the quantum computer has made the problem manageable by the classical computer. The classical computer's output is unaffected by quantum noise and is generally more readily available than that of the quantum computer. By identifying when the quantum computer has generated enough information for it to "hand over" the task of completing the computation to the classical computer, the operator can not only perform computations traditionally considered difficult for classical computers, but also avoid noise in the final answer (resulting in a more accurate final result), freeing up limited quantum computing resources for other computations that classical computers still struggle with, among other benefits.

[0008] Hamiltonian calculation Specifically, relative to computing the characteristic solution of the Hamiltonian, this disclosure allows control of the computation to be transferred back to the classical computing system, by using a classical computing system to construct the initial complete Hamiltonian, using a quantum computing system to generate the corresponding wavefunction, and using the classical computing system to determine how and when to generate the dimensionality-reduced version of the complete Hamiltonian (referred to herein as the subspace Hamiltonian). By controlling when and how control is transferred back to the classical computing system (or operations are transferred back to the quantum computing system), this disclosure provides an output that mitigates the inherent uncertainty of quantum noise in quantum computing systems for Hamiltonian computation. Furthermore, compared to other hybrid computing methods, the described method uses fewer computational resources and provides operators with various control mechanisms to select different accuracy thresholds, resource allocations, and computational profiles, as well as other benefits and technical improvements that will become apparent to those skilled in the art upon a detailed review of this disclosure.

[0009] Hamiltonian representation The Hamiltonian can be represented as a matrix, which simplifies quantum chemistry problems to matrix eigenvalue problems. The size of such matrices grows exponentially with respect to the number of electrons and orbitals. The target state satisfying this eigenvalue problem is typically very sparse, meaning that a large portion of the basis states constituting the state have amplitudes (or contributions) that are zero or negligible. Therefore, one can use basis states that make non-negligible contributions to the target state to perform a dimensionality reduction of the Hamiltonian matrix into a smaller, denser matrix. Such a compressed matrix may still have an exact eigenvalue or an approximate eigenvalue.

[0010] Solution extension The problem of finding eigenvalues ​​for sparse matrices can be extended beyond chemical problems. Any optimization problem that can be formulated as a quadratic unconstrained binary optimization problem can be compressed to a certain degree of accuracy. In this case, the ability to solve large matrices can stem from efficient methods of compressing large matrices into smaller, representative matrices.

[0011] All aspects of this disclosure One aspect of this disclosure provides a method that uses a hybrid computing system to output the ground state value and ground state energy value of a chemical system calculated by both quantum computing devices and classical computing devices. This method is achieved by performing the following steps via the quantum computing system: using a test wavefunction preparation protocol, for a system with 2... n The wavefunction is prepared from the complete Hamiltonian of the chemical system with basis states, such that the wavefunction facilitates the sampling of basis states that contribute non-negligiblely to the characteristic solutions of the complete Hamiltonian; once the quantum computing system has prepared a sufficiently accurate wavefunction, the iterative calculation is terminated; via the quantum computing system, based on the compound occupancy... KThe probabilities of each basis vector state are sampled from the wavefunction prepared in the last iteration of the iterative calculation. N 1 sample to describe 2 n In each of the basis states N A subset of basis states; via a classical computer system, using a selection protocol to... N Selecting from the basis states K A subset of basis vector states; a subspace Hamiltonian of the complete Hamiltonian is constructed using a classical computer system; and the classically calculated ground state characteristic solution of the complete Hamiltonian is calculated and output based on the characteristic solution of the classically calculated subspace Hamiltonian using a classical computer system.

[0012] Cross-iteration change concept In the aforementioned method, the initial state preparation protocol can include various protocols that vary across different iterations, the experimental wavefunction preparation protocol can include various protocols that vary across different iterations, different stopping criteria can be used across various iterative loops, and a single loop can exit under different conditions in various iterations, for use from 2 n Selecting from the basis states K The selection protocol for each basis vector state to be included in the Hamiltonian of the subspace can include a variety of different protocols derived across different iterations.

[0013] Additional aspects Additional aspects, features, and advantages of the disclosed methods and apparatus are described in the following detailed description and accompanying drawings, and will be apparent therefrom. The features and advantages described herein are not exhaustive, and in particular, many additional features and advantages will be apparent to those skilled in the art, taking into account the drawings and description. Furthermore, it should be noted that the language used in the specification has been chosen primarily for readability and guidance purposes, and not to limit the scope of the subject matter of the invention. Attached Figure Description

[0014] Figure 1 An example hybrid computing system according to an embodiment of the present disclosure is illustrated.

[0015] Figures 2A to 2C Illustrations of various computational spaces according to embodiments of this disclosure are provided.

[0016] Figures 3A to 3J This is a flowchart of a selection protocol for selecting which basis states in the complete Hamiltonian to include in the subspace Hamiltonian, according to an embodiment of this disclosure.

[0017] Figures 4A to 4J Example diagrams illustrating the implementation of certain selection agreements according to embodiments of this disclosure are provided.

[0018] Figures 5A to 5C This is a flowchart of an example method for improving the computational efficiency of a hybrid quantum-classical computing system when determining characteristic solutions of a Hamiltonian, according to an embodiment of the present disclosure, wherein the Hamiltonian can be used to represent an element or compound.

[0019] Figure 6 This is a flowchart of an example method for improving the computational efficiency of a hybrid quantum-classical computing system when using Pauli sampling or Pauli measurement to determine characteristic solutions of a Hamiltonian, according to an embodiment of the present disclosure, wherein the Hamiltonian can be used to represent an element or compound.

[0020] Figure 7 This is a flowchart of an example method for improving the computational efficiency of a hybrid quantum-classical computing system when determining characteristic solutions of a Hamiltonian without using Pauli sampling or Pauli measurement, according to embodiments of the present disclosure, wherein the Hamiltonian can be used to represent an element or compound.

[0021] Figure 8 This is a flowchart of an example method for improving the computational efficiency of a hybrid quantum-classical computing system according to an embodiment of this disclosure, as can be used to represent elements or compounds.

[0022] Figure 9 This is a flowchart of an example method for improving the computational efficiency of a hybrid quantum-classical computing system according to an embodiment of this disclosure, as can be used to represent elements or compounds.

[0023] Figures 10A to 10B This is a flowchart of an example method for improving the computational efficiency of a hybrid quantum-classical computing system according to an embodiment of this disclosure, as can be used to represent elements or compounds.

[0024] Figures 11A to 11F Examples are given regarding the construction of subspace Hamiltonians according to embodiments of this disclosure, for different... K Various performance improvements to the value.

[0025] Figure 12 A classic computing device according to an embodiment of this disclosure is illustrated.

[0026] Figure 13 A quantum computing device according to an embodiment of the present disclosure is illustrated.

[0027] The examples given herein illustrate certain non-limiting embodiments in one form, and such examples should not be construed as limiting the scope of the appended claims in any way. Detailed Implementation

[0028] Examples and Implementation Schemes The subject matter of this disclosure will now be described and discussed in more detail with reference to the accompanying drawings, which illustrate some (but not all) embodiments of the invention. Unless otherwise stated, the same reference numerals refer to the same elements or parts throughout the text. The subject matter of this disclosure may be embodied in many different forms and should not be construed as limited to the specific embodiments set forth herein. Rather, these embodiments are provided to enable this disclosure to meet applicable legal requirements. Indeed, many modifications and other embodiments of the subject matter of this disclosure will readily occur to those skilled in the art. Therefore, it should be understood that the subject matter of this disclosure is not limited to the specific embodiments disclosed, and that modifications and other embodiments are intended to be included within the scope of the appended claims.

[0029] Chemical system The term "chemical system" can refer to any composition or collection of atoms that can be represented by a chemical formula, including molecules, ions, free radicals, etc., in which the atoms are bonded together by covalent or non-covalent bonds, including homonuclear or heteronuclear collections.

[0030] Quantum representation of chemical systems When representing chemical systems in a quantum computer, qubits are used to represent the properties of the chemical system. Each qubit is based on a two-level quantum system that encodes either a computational basis state of |0〉 or |1〉, but can simultaneously be a coherent superposition of these two values ​​before being measured. Therefore, using n A quantum computer with 100 qubits has 200 qubits that can be used to describe this chemical system. n Two basis vector states. n Each of the basis states is associated with the probability amplitude α. i Relatedly, the square of this probability value represents the probability that the chemical system is in a given basis state, where α i The value is determined by sampling the quantum representation of the chemical system and recording the frequencies at which the chemical system is found to be in a given basis state. For example, if X samples are taken from a qubit system (e.g., n=1), it may be determined that Y% of the time is spent in the computed basis state |0> (e.g., ...). Y = | a 0| 2 * 100), and there is Z% of the time in |1> (e.g., Z = | a 1| 2 * 100), and the system can be represented as a wave function | >= α 0|0〉+ α 1|1〉.

[0031] Hamiltonian ( H ) The Hamiltonian is an operator in quantum mechanics. For a chemical system, it represents the total energy and provides the possible energies based on a set of eigenvalues ​​and eigenvectors. The energy of a given state in which the system can be is given by the expected value of the Hamiltonian relative to that state.

[0032] Types of Hamiltonian The complete Hamiltonian includes the value of each basis state of the chemical system it represents; however, many of these basis states contribute zero or close to zero to the characteristic solution, indicating a low probability that the chemical system is in that basis state when it is measured. Therefore, the number of basis states in which the chemical system is in a high probability is relatively small compared to the total number of basis states. Using this sparsity, a subspace Hamiltonian can be constructed using only a small number of basis states associated with the characteristic solution of the complete Hamiltonian. If the basis states used to construct the subspace Hamiltonian match those that best represent the characteristic solution of the complete Hamiltonian of the chemical system, then the characteristic solution of the subspace Hamiltonian can approximate the characteristic solution of the complete Hamiltonian of the chemical system, while reducing the computational complexity of the representation.

[0033] The meaning of identifying the Hamiltonian of the representative subspace The computation of the characteristic vector amplitudes of the full Hamiltonian of a chemical system is typically recommended to be performed via a quantum computer. This is because the computational complexity of chemical systems increases with size, making solutions derived via classical computers or human thought processes intractable. Operators can use various quantum computing methods, such as variational quantum characteristic solvers (VQEs), to search for the ground state of the chemical system from the initial state using ansatz (ansatz can make best guesses about the final ground state using various methods known to those skilled in the art). VQEs are iterative processes, which consume significant computational resources on a quantum computer and may end with iterations producing suboptimal solutions that do not correspond to the ground state (e.g., getting stuck in local minima instead of global minima), further wasting limited quantum computing resources. However, by identifying when the quantum computer has generated a sufficiently accurate wavefunction to produce a representative subspace Hamiltonian (typically before it arrives at a quantitatively complete answer on its own), and as long as the subspace Hamiltonian sufficiently shrinks the problem space to make the computation tractable at the classical level, the quantum computer can hand over further computation to a classical computer to solve the remaining computations. Therefore, hybrid computing systems can use a transfer procedure to transfer computation (using the full Hamiltonian) from a quantum computer to a classical computing system (using the corresponding subspace Hamiltonian) to complete the computation, thereby saving limited quantum computing resources, while also producing outputs whose accuracy is unaffected by quantum background noise, as well as other benefits of using hybrid computing systems.

[0034] Hybrid computing system Figure 1 An example hybrid computing system 100 according to an embodiment of this disclosure is illustrated. Hybrid computing system 100 includes a classical computing system 110 that interfaces with a quantum computing system 120 to perform computations using both classical and quantum computing techniques. As used herein, hybrid computing system 100 refers to a combination of classical and quantum computers, rather than a combination of analog and digital computers (the latter having been referred to differently in other fields as "hybrid computers"). Each of the classical computing system 110 and the quantum computing system 120 may include one or more computing devices and various communication interfaces. Regarding... Figure 12 Example classical computing system 110 is discussed, and regarding Figure 13 Example quantum computing system 120 is discussed.

[0035] Benefits of Hybrid Computing Systems Traditionally, classical computing system 110 has been used to define sequences of quantum gates or quantum circuits for quantum computing system 120 and to receive the results of quantum computations performed by quantum computing system 120. In contrast, hybrid computing system 100 allows for the distribution of workloads among systems based on their relative strengths in certain tasks, thereby efficiently performing both classical and quantum computing. This results in improvements in overall system performance, reductions in computational resources required by traditional computing systems, increased speed and accuracy of computations performed, and other technological improvements and benefits.

[0036] Various architectures of hybrid computing systems In various implementation schemes, the hybrid computing system 100 can be organized according to various architectures, including batch quantum computing architecture, interactive quantum computing architecture, integrated quantum computing architecture, and distributed quantum computing architecture.

[0037] Batch processing computing architecture In a batch quantum computing architecture, classical computing system 110 defines quantum circuits and submits these quantum circuits as jobs to the quantum processing unit (QPU) of quantum computing system 120, which returns the results to classical computing system 110. However, batch processing multiple quantum circuits into a single job reduces the waiting time between submissions, thus allowing hybrid computing system 100 to run multiple jobs more quickly. Therefore, when quantum computing system 120 completes one quantum circuit "job," the next quantum circuit "job" is ready for analysis.

[0038] Interactive quantum computing architecture In interactive quantum computing architectures, operators can specify that quantum circuits be executed repeatedly using different (or the same) parameters. Jobs can be logically grouped into a session and given priority over non-session jobs. While sessions allow for shorter queuing times and longer runtime problems, quantum bit states do not persist between iterations. Examples of problems where this approach can be used include Variable Quantum Characteristic Solvers (VQEs) and Quantum Approximation Optimization Algorithms (QAOAs).

[0039] Integrated quantum computing architecture In the integrated quantum computing architecture, classical computation is performed while the physical qubits are in coherent states, as the classical computing system 110 and the quantum computing system 120 operate collaboratively. Although potentially limited by qubit lifetime and error correction, the integrated architecture allows quantum programs to include general-purpose programming tools beyond simple circuit analysis (e.g., loops, nested conditional statements, etc.), which can use one or more states from the quantum circuit as values ​​for variables. Advantageously, the classical computing system 110 can allow for various qubit reuse techniques; thus allowing the quantum computing system 120 to have fewer physical qubits but provide a large number of "virtual" qubits to run more complex computations.

[0040] Distributed quantum computing architecture In a distributed quantum computing architecture, classical computing system 110 works in conjunction with quantum computing system 120 using logical qubits. The long coherence time provided by the logical qubits enables complex and distributed computations across cloud computing environments. Therefore, various computing devices can reside in different environments and can be shared with other users on a Platform as a Service (PaaS) usage model.

[0041] Example computation space Figures 2A to 2C Illustrations of various computational spaces according to embodiments of this disclosure are provided. Figures 2A to 2B An example is given showing the representation of the electronic Hamiltonian of a hydrogen molecule (H2), in which Figure 2A It is the matrix representation of the complete Hamiltonian of H2, and Figure 2B Example matrix representations of the subspace Hamiltonian of H2 are shown. Figure 2C Various computational spaces relative to the complete basis space (from which the complete Hamiltonian can be computed) are illustrated.

[0042] Hamiltonian matrix The Hamiltonian can be represented by a matrix, making it possible to use quantum computing devices on a quantum computer. n When there are 100 qubits, the dimension of the Hamiltonian matrix can be described as 2. n× 2 n For an H2 molecule using 4 qubits, the molecule can be completely represented in a 16 × 16 Hamiltonian matrix, such as... Figure 2A As shown.

[0043] H 2 Hamiltonian matrix Figure 2A The values ​​shown in the matrix represent matrix elements in each of the different basis states, where each position in the matrix represents a matrix element. ij Corresponding to the basis state b i and b j The matrix elements. It is obvious that the characteristic solution of the complete Hamiltonian can be well approximated by the sparse solution; many of these basis states have amplitudes of zero or close to zero, which indicates that the contribution to the characteristic solution of the complete Hamiltonian is zero or close to zero.

[0044] The linear sum of Pauli words As will be understood, this matrix can also be represented as a complex linear summation of Pauli words, which is more memory efficient because the complexity of the matrix grows exponentially with respect to the number of qubits. As those skilled in the art will understand, the complex linear summation representation of the complete Hamiltonian is precisely the object of VQE operations because this format is memory efficient (although both encode the same data), although various black-box representations with element-wise access protocols can also be used to further improve efficiency.

[0045] Subspace Hamiltonian To reduce the computational complexity of performing mathematical operations on the full Hamiltonian, subspace Hamiltonians were created. A subspace Hamiltonian is a reduced-dimensional matrix whose eigenvalues ​​approximate the eigenvalues ​​of the full Hamiltonian, thus allowing classical computers to solve for approximate eigenvalues ​​of the full Hamiltonian, a task typically considered intractable on classical devices due to the exponential size of the matrix. Because of the sparse nature of the approximation of the full Hamiltonian's eigenvalues ​​(having many amplitudes of zero or near zero), the influence of basis states with zero or near-zero amplitudes on the computation of approximate eigenvalues ​​of the full Hamiltonian is negligible. Therefore, by using important basis states (i.e., basis states with non-negligible amplitudes in the eigenvalues) to construct the subspace Hamiltonian, it provides a highly accurate approximation of the full Hamiltonian, making classical computation highly accurate in terms of energy consumption.

[0046] Basis state selection To create the subspace Hamiltonian, various selection criteria were used to identify the values ​​210a-d (usually or collectively referred to as value 210) used to construct the subspace Hamiltonian. This was achieved via a 2... n Appropriate selection from the basis vector states K A number of basis states can reduce the complexity of mathematical operations performed on the Hamiltonian of the subspace to a certain extent, allowing the quantum computing device 1300 to offload further computations to the classical computing device 1200, which would otherwise be difficult to handle on the classical computing device, without making a significant sacrifice in the accuracy of the final result. Basis states are selected according to selection criteria, such as symmetry criteria (e.g., checking whether the number of electrons is conserved), screening criteria (e.g., values ​​above a given threshold), overlap criteria (e.g., identifying overlap between basis states and other basis states, and identifying important groups of basis states); or randomized selection, and combinations thereof.

[0047] Example subspace Hamiltonian The sampling and application selection protocol provides crucial eigenvalues ​​for the complete Hamiltonian. K After the basis states, create a system with the selected value 210. K × K Hamiltonian of a subspace with reduced dimensionality ' ,like Figure 2B As shown, where K = 2. Subspace Hamiltonians are used from complete Hamiltonians. K The matrix is ​​constructed using the selected basis vectors from each sample. The selected values ​​are arranged as follows: K × K A matrix (or represented as a linear sum of the corresponding Pauli terms or other efficient representation) can be arranged in various orders. As will be understood, different selected basis states and different... K The Hamiltonian of other subspaces will produce a value other than Figure 2B The example provides subspace Hamiltonians other than those provided in the subspace Hamiltonian.

[0048] Changes in computation space Figure 2C Examples are given except Figures 2A to 2B The example space is exemplified in the example space. Various different computational spaces exist beyond this. In seeking specific characteristic solutions for the Hamiltonian, the Hilbert space of all possible states that the system might occupy must be searched. Complete basis space 220 ( N This includes all the 2 quantum devices that can be prepared for the system being analyzed. nThere are 100 possible basis states, and several potential subspaces can be further identified within the complete basis space 220. The symmetry space 230 (S) includes all these basis states that the system can efficiently occupy (e.g., based on system properties such as electron number and total spin), and excludes those basis states from the complete basis space 220 that are not considered efficient for the system under analysis; typically, the symmetry space is smaller than the complete basis space (e.g., S < 1). N The proposed space is 240 ( A The proposed space 240 contains all these basis states that can be sampled from the states prepared by the proposed quantum device, excluding basis states in the full basis space 220 that cannot be accessed by states prepared for the test wavefunction. Additionally, depending on the proposed structure, the proposed space 240 may contain basis states that are not considered valid for the system (e.g., if the proposed space does not maintain electron number conservation), such that some basis states in the proposed space 240 lie outside the symmetry space 230. Therefore, in some embodiments, the proposed space 230 is contained within the symmetry space 230, while in other embodiments, the proposed space 230 extends beyond the symmetry space 230.

[0049] Optimal basis state When calculating the characteristic solution of a Hamiltonian, not all basis states provide equal values ​​as input; some basis states are more important for the accuracy of the final result. For example, the maximum optimal basis space 250 includes all basis states associated with the target characteristic solution, while the minimum optimal basis space 260 includes the smallest set of basis states from the optimal basis space 250 necessary to achieve the required accuracy level for the characteristic solution (e.g., in a chemical system, this would be chemical accuracy; e.g., 1.6 × 10⁻⁶). -3 Ha). There exist various subspaces of size between the maximum optimal basis space 250 and the minimum optimal basis space 260. The prior space 280 includes all basis states that are a priori known to be in the maximum optimal basis space 250, which can be known from previous analyses of the system being analyzed.

[0050] Choose space As used in this article, the core space is 270 ( C ) refers to the selection of space (through a selection protocol) A A set selected from ) K There are 260 basis states. Ideally, the core space 270 would match the minimum optimal basis space 260, but limitations imposed by available computational resources may affect the operator's ability to achieve this state for the core space 270. Furthermore, when initially considering the problem at hand, one might not know exactly which basis states in the complete basis space 220 belong to the minimum optimal space 260, and due to the complete basis space (260, ...n The number of states in the space is exponential, and neither random selection nor brute-force selection is a feasible method for identifying the basis states that constitute the minimum optimal space 260.

[0051] Framework Introduction Therefore, this disclosure provides a framework for efficiently obtaining basis states in a minimum optimal space 260 (e.g., to optimize the selection of the core space 270) by sequentially applying two different mechanisms. A hypothesis on the quantum device is used to prepare states using a subset of basis states in the full basis space (i.e., the hypothesis space 240), which serves as the search region for a given iteration of the framework. Depending on how the hypothesis is constructed, the hypothesis space 240 may overlap with the minimum optimal space 260 to varying degrees, or even not overlap at all. Therefore, by means of a trial wavefunction preparation protocol, the framework aims to maximize the region of the minimum optimal space 260 included in the hypothesis space 240 by varying the states prepared on the quantum device.

[0052] Select the application of the protocol After iterative attempts to optimize the overlap between the proposed space 240 and the minimum optimal space 260, a selection protocol is applied to the basis states in the proposed space 240 to further narrow the search region to a core space 270, which includes only states in the overlap between the proposed space 240 and the minimum optimal space 260. This reduction in the number of basis states facilitates the application of classical computer systems in finding characteristic solutions to the Hamiltonian. In most cases, the proposed space 240 will contain states not included in the minimum optimal space 260, which are filtered out to produce the core space 270. Ideally, in the case where the experimental wavefunction preparation protocol prepares perfectly sampled states on a quantum device, there will be no reduction in the number of basis states in this step, as the proposed space 240 would be equivalent to the minimum optimal space 260; however, this is unlikely to occur in practice.

[0053] Proposed spatially overlapping targets By sampling the hypothetical space 240, a core space 270 is generated, which avoids exponential scaling when selecting the basis states to be manipulated. A corollary of this sampling is that only basis states contained within the hypothetical space 240 can form part of the core space 270. If a large portion of the minimum optimal space 260 lies outside the hypothetical space 240, the core space 270 may not represent the minimum optimal space 260, and the hypothetical space 240 alone cannot provide these missing basis states. One way to improve the generation of the core space 270 is to prepare quantum states that have greater overlap with the target characteristic solution to better overlap with the minimum optimal space 260. Another way to improve the generation of the core space 270 is to supplement the hypothetical space 240 with states obtained through some other procedure (e.g., a second hypothetical space, or a priori space 280 from earlier analysis).

[0054] The proposed space is overlaid using basis states known to be part of the minimum optimal space. Furthermore, the use of the prior space 280 facilitates the experimental wavefunction preparation protocol in creating a representative or sufficiently overlapping hypothetical space 240. States contained in the prior space 280 are known to be included in the minimum optimal space 260 and can be obtained via various preprocessing techniques (such as classical ab initio computation) or from previous iterations of the described framework. These states guarantee that the hypothetical space 240 has a non-zero minimum overlap with the minimum optimal space 260, which helps the experimental wavefunction preparation protocol prepare quantum states capable of generating a hypothetical space 240 with greater overlap to the minimum optimal space 260.

[0055] Explore the proposed space using basis states that are known to be part of the minimum optimal space. In addition to using the prior space 280 to guide the overlap of the proposed space 240 to include the values ​​in the prior space 280, the prior space 280 can also be used to assist in the experimental wave function preparation protocol to prepare quantum states that produce a proposed space 240 that does not overlap with (or attempts to reduce the overlap with) the prior space 280, in order to explore different parts of the minimum optimal space 260 that are explicitly not included in the prior space 280.

[0056] Considerations for generating the core space The core space 270 is the result of applying a selection protocol to the states in the proposed space 240, and the relevance of the core space 270 is related to the effectiveness of the selection protocol. If the selection protocol includes states from the proposed space 240 that are not in the minimum optimal space 260, or if the selection protocol excludes states from the proposed space 240 that are in the minimum optimal space 260, then the selection protocol is not optimal. In the first case, the efficiency of the classical feature solver program will decrease (potentially to the point of being infeasible on classical computing systems). In the second case, the classical feature solver program cannot achieve the accuracy required for the feature solution.

[0057] Idealized selection protocol Ideally, the selection protocol extracts only those states from the proposed space 240 that are also in the minimum optimal space 260. Therefore, the required level of efficiency for the selection protocol is proportional to the quality of the proposed space 240 (e.g., the extent to which the proposed space 240 contains only basis states of the minimum optimal space 260). If the proposed space 240 equals the minimum optimal space 260, the selection protocol only needs to select all basis states in the proposed space 240. However, if the proposed space 240 contains basis states that are not in the minimum optimal space 260, the selection protocol must be able to filter these basis states from the proposed space 240 when defining the core space 270, so as to retain only those basis states that are also in the minimum optimal space 260.

[0058] Some variable properties of space Figure 2C The hypothetical space 240, core space 270, and extended space 290 are illustrated with dashed lines because the relative size and position of these spaces with respect to the complete basis space 220 may vary depending on different computational settings. For example, depending on how the hypothetical or experimental wavefunction is generated to represent the complete basis space 220, the contents of the first hypothetical space 240 may differ from the contents of the second hypothetical space 240 within the same complete basis space 220. Similarly, even within the same hypothetical space 240, the core space 270 may vary depending on how the basis states from the hypothetical space 240 are chosen.

[0059] Expansion of core space In various implementations, the core space 270 can be extended via various methods by adding an extension space 290(X) to the core space 270. For example, two or more core spaces 270 selected via different selection protocols or from different proposed spaces 240 can be merged via a union space of multiple core spaces 270, where one core space 270 acts as an extension space 290 of another core space 270. Alternatively, the prior space 280 can be used as an extension space 290. Other extension spaces 290 can be identified by Hamming distance (e.g., Hamming space), a basis set of α / β exchange, or other spaces or basis sets located within the symmetric space 230 but outside the current proposed space 240 and selected by methods now known or future developed, and various combinations thereof. A combination of one or more extension spaces 290 with the initial core space 270 discovered from the proposed space 240 can be referred to as the “extended core space”.

[0060] Hanming Space Introduction Hamming space is based on each basis vector state b i The Hamming distance is defined in the remaining space of the symmetric space 230 (e.g., the portion of the symmetric space 230 that is not overlapped by the proposed space 240), and this Hamming distance identifier b i The number of positions that differ from Hartree-Fock (HF) or other basis states. For example, a computational system can construct a probability distribution based on Hamming distance by assigning weights wi to each bit string bi. Then, the weights are normalized to obtain the probability p of sampling the bit string bi. bi From this, M basis states of the Hamming space are identified. These M basis states can form an extension space 290, which is added to the core space 270 as is, or a subset of these M basis states can be identified and added to the core space 270 through various selection protocols.

[0061] α / β Exchange introduction Each basis state of a chemical system can be represented as a string of bits (e.g., a bit string), where the value of each bit corresponds to the presence or absence of an electron in the corresponding orbital (1 indicates presence, 0 indicates absence). For example, a chemical system with two α electrons, four α orbitals, two β electrons, and four β orbitals can be represented by an eight-bit bit string, where the first half of the bit string corresponds to the α orbitals and the second half corresponds to the β orbitals (e.g., represented as ααααββββ), but those skilled in the art will understand that other representations can also be used. Thus, the basis state of the system represented as "11110000" in core space 270 will have α bits "1111" and β bits "0000", and when an α / β swap is performed, the basis state "00001111" will be selected for use in extension space 290. These α / β exchanged basis states can form an extension space 290, which is added to the core space 270 as is, or a subset of the α / β exchanged basis states can be identified and added to the core space 270 via various selection protocols.

[0062] Core space compression In addition to expanding the core space 270, the computing system can also compress the core space 270 (such as after adding the expansion space 290 to the core space 270) to maintain the number of basis states defining the core space. K This compression can be performed by identifying various basis states that have little or no effect on the calculated energy of the chemical system, thereby reducing the number of basis states used to compute characteristic solutions and various derived values, thus saving computational resources, maintaining the number of desired states used for analysis in a given computation, and their combinations.

[0063] choose K Value factors Because the operator selects the number of basis states used to compute the characteristic solution. K There are no unlimited permissions when choosing a value, so certain factors can be used to select the appropriate one. K These factors include, but are not limited to: (1) the probability derived for the quantum state being sampled, (2) the available computational resources of the quantum computer, (3) the available computational resources of the classical computer, and (4) the value in case the previously chosen value is insufficient to compute the characteristic solution. K Prior values ​​(e.g., to ensure that a larger value is chosen in subsequent iterations). K value).

[0064] Sample value The first factor to consider is the amplitude of the basis states in the quantum state being prepared. The amplitude of the quantum state (α) i The sampling number is given. iThe probability of each basis state | α i | 2 For example, if the operator intends to take samples K desired There are several states, but the number of basis states (j) with non-negligible sampling probabilities in the quantum state is less than [a certain number]. K desired (For example, j < K desired ),but K actual The maximum value is j or smaller (e.g., K actual ≤ j). This can occur if the transfer from the quantum computer to the classical computer is performed too early, and the magnitudes of the basis states in the quantum state are not qualitatively accurate before the state is sampled. In practice, noise on the quantum computer may cancel out this effect and allow a sufficient number of basis states to be sampled. However, this cancellation does not mean that the algorithm can terminate earlier on noisy devices, because there is no reason to believe that the basis states introduced by noise will contribute non-zero value to the final characteristic solution.

[0065] Available quantum computing resources The next factor is the available quantum computing resources, which includes the number of samples an operator can undertake when sampling basis states from a quantum state. An operator has a finite number of attempts to sample basis states (e.g., based on time, processor cycles, and the time to hold values ​​in a qubit), and if the operator samples only a quantity j (where j < ... K desired If the basis vectors of ) are , then the operator at j = K actual There will be no other choice regarding size.

[0066] Operator to K Selected input The two factors mentioned above do not necessarily limit K desired The size of the sample is important. For example, the operator can always guess which samples should be non-zero in the final solution, or provide values ​​from an external source (or previous runs of the program). However, both of these factors do impose a hard limit on the number of basis states that can be sampled directly from a quantum state / iteration of the process, since quantum computing devices (and access to them) may only provide a given number of results.

[0067] Available classical computing resources The third factor is the available classical computing resources. These resources have two limitations: the time and space (memory) available to classical computers. K The value of needs to be small enough that a classical computer has the resources required to compute the characteristic solutions of the Hamiltonian in the subspace. For example, if all possible basis states are to be used, the classical computation time and space required to find the characteristic solutions of the Hamiltonian grow exponentially with the problem size, so the resources required by a classical computer may make it practically impossible to compute even for a medium-sized chemical system on the most powerful classical computing system.

[0068] K Prior values The fourth factor provides a balance: on the one hand, the third factor forces the selection of lower-ranking individuals. K desired On the one hand, there is pressure regarding the value, and on the other hand, there is a desire for the subspace Hamiltonian to be more similar to the complete Hamiltonian, thus allowing for the selection of higher values. K desired The expected value. For example, if in an earlier iteration it was... K If the selected value is determined to be too low to create a representative subspace Hamiltonian, the system will ensure that the next iteration will not be a Hamiltonian. K Choose the same or lower value. Conversely, if it was in an earlier iteration... K If the chosen value is still computationally intractable for classical computing systems, the system will ensure that the next iteration will not be [value missing]. K Choose the same or a higher value. Therefore, the hybrid system may initially try a lower value in an earlier iteration of the handover procedure. K desired The value is initially set, but when the evaluation result of the resulting subspace Hamiltonian fails to represent the complete Hamiltonian, it continuously increases in subsequent iterations. K desired Value. By implementing the rule according to the fourth factor, the hybrid system can gradually adjust the computational load allocated to the classical computing system until a representative subspace Hamiltonian is produced. As will be understood, compared to earlier iterations, the adjustment... K The magnitude of the value can increase or decrease the number of basis states included in the subspace Hamiltonian; thus allowing the operator to fine-tune the computational load and accuracy of the solution generated by the classical computing system.

[0069] Select Protocol In addition to being able to adjust the number of basis states used to create the subspace Hamiltonian across several iterations (e.g., choosing different ones in different iterations), K In addition to the benefits provided by the value, this disclosure also provides a variety of possibilities for selection protocols to determine which basis vector states are selected from the complete Hamiltonian as subspace Hamiltonians.K There are basis vector states. Therefore, by using different selection protocols in different iterations, even if the same basis vector is selected for the same complete Hamiltonian, the same basis vector state can be obtained. K The value can also generate multiple different subspace Hamiltonians, which may be helpful in determining which basis vectors in the complete Hamiltonian represent the most salient values ​​for the analysis. K When considering each basis vector state, different aspects are prioritized.

[0070] Method for selecting basis states from the whole Hamiltonian to be included in the subspace Hamiltonian Figures 3A to 3J This is a flowchart of selection protocols 300a-300j (generally or collectively referred to as selection protocol 300) for selecting which basis vector states in the complete Hamiltonian to include in the subspace Hamiltonian, according to an embodiment of this disclosure. Figures 4A to 4H Example figures 405a-h (generally or collectively referred to as Figure 405) are provided for implementing certain selection protocols. In Figure 405, nodes 415a-t (generally or collectively referred to as node 415) each represent a sampled basis state, and selected nodes 415 are represented by solid circles, while unselected nodes are represented by hollow circles.

[0071] Electronic Conservation Selection Protocol Figure 3A This is a flowchart of a method 300a for implementing an electron conservation selection protocol according to an embodiment of this disclosure. The electron conservation selection protocol samples from a quantum device. N Selecting from the basis states K a basis state (e.g., as a subset of available basis states, where K ≤ N ), where represents the group K The number of "1"s in the bit string of each selected basis state is the same as the number of electrons considered in the problem. In other words, method 300a applies a symmetry criterion to select the sampled basis states to be included in the subspace Hamiltonian from the complete Hamiltonian, where the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to the number of samples to be included in the subspace Hamiltonian. K These values ​​are represented by a bit string having a number of "1"s equal to the number of electrons in a chemical system represented by a complete Hamiltonian.

[0072] box 310a- Sampling from quantum devices N basis states At box 310a, the quantum computing system samples the experimental wavefunction to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | a j | 2 .

[0073] box 320a - Removed items that did not contain the corresponding number "1” bit string At box 320a, if the number of "1"s contained in the bit string of the basis vector state does not match the number of electrons in the chemical system being analyzed, the classical computational system removes it from consideration as included in the subspace Hamiltonian. As used herein, the number of bit strings determined to have the same number of "1"s as the number of electrons is called E. BS For example, in a chemical system with two electrons, the sampled... N The basis states are chosen from the given basis states, specifically the bit string "1010" because the number of "1"s equals the number of electrons. However, the basis state with the bit string "0111" is not chosen because it contains three "1"s, while the problem only involves two electrons. Each bit string with the corresponding number of "1"s will be available as a basis state. K A portion of the chosen basis states, and E BS The value will be six (e.g., 0011, 0101, 0110, 1001, 1010, and 1100 for two electrons). Therefore, of the sixteen bit strings sampled for an exhaustive sampling of a two-electron system (e.g., 0000 to 1111), ten bit strings will be removed; leaving six bit strings as... K Select a portion (e.g., E) BS = 6).

[0074] box 330a - Supplementary options / Remove At box 330a, from E, which is identified as satisfying the selection protocol BS The basis states are selected from the basis states to be used in calculating the characteristic solutions. When E BS = K (or when E) BS =K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when E... BS > K At that time, a supplementary selection process can be performed to identify E. BS A subset of basis states is used to reduce the selection to a subset for the subspace Hamiltonian. K Each of the following is a basis vector state. Similarly, when E... BS < K At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states in each basis state are used for the subspace Hamiltonian (e.g., K remainder = K desired - E BSTypically, to ensure that a set is ultimately selected... K desired Each basis state can be selected as one or more sets. K supplemental Add a basis state to the initial quantity K selected basis vector states (of which K supplemental Each of the basis states is preferably not the initial one. K selected (members of each basis state), or can be obtained from K selected Select one or more sets from the basis vector states. K removal Each basis vector state, from the initial quantity K selected Members are removed from each basis state. Once the Hamiltonian for constructing the subspace is determined... K desired The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0075] α-β Conservative Choice Protocol Figure 3B This is a flowchart of a method 300b for implementing an α-β electron conservation selection protocol according to an embodiment of this disclosure. The α-β conservation selection protocol samples from a quantum device. N Selecting from the basis states K a basis state (e.g., as a subset of available basis states, where K ≤ N ), which indicates K The number of "1"s in the bit string of each chosen basis state at the α orbital position is the same as the number of α electrons in the problem, and represents... K The number of "1"s in the bit string of each selected basis state at the β orbital position is the same as the number of β electrons in the problem. In other words, method 300b applies a symmetry criterion to select the sampled basis states to be included in the subspace Hamiltonian from the complete Hamiltonian, where the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to the number of samples in the subspace Hamiltonian. K These values ​​are represented by a bit string that has the number of "1"s at positions corresponding to a given type of orbital equal to the number of electrons in that given type of orbital in the chemical system, as represented by the full Hamiltonian.

[0076] box 310b - Sampling from quantum devices N basis states At box 310b, the quantum computing system samples the experimental wavefunction to obtain a value from 2... nA set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 .

[0077] box 320b - Remove items that do not contain the corresponding number of items in the corresponding orbital. "1” bit string At box 320b, if the number of "1"s in the bit string of the basis vector state at the positions corresponding to the α and β orbitals in the chemical system being analyzed does not match the number of electrons, the classical computational system removes it from consideration as included in the subspace Hamiltonian. As used herein, the number of electrons in the α orbital is denoted as E. α The number of electrons in the β orbital is designated as E. β For example, for a electron with two α electrons (e.g., Eα), α =2), 4 α orbitals, and two β electrons (e.g., E2). β The problem of having 2 α orbitals and 4 β orbitals can be addressed by representing the chemical system using an octet of bits, where the first half of the bit string corresponds to the α orbital and the second half corresponds to the β orbital (e.g., represented as ααααββββ), though those skilled in the art will understand that other representations are also possible. Continuing the example, a classical computational system would remove any basis states having bit strings with fewer or more than two "1"s in either the α or β portion (e.g., 11101100, which has three "1"s in the α portion and two "1"s in the β portion; 10101000, which has two "1"s in the α portion and one "1" in the β portion, etc.). In contrast, a classical computational system preserves basis states with two "1"s in both the α and β portions (e.g., 11001010 and 1011001) for use with... K A selected basis state. Having satisfying E α and E β The total number of basis states of the bit strings of both can be understood as E BS .

[0078] box 330b - Supplementary options / Remove At box 330b, from E, which is identified as satisfying the selection protocol BS The basis states are selected from the basis states to be used in calculating the characteristic solutions. When E BS = K (or when E) BS = K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when E... BS> K At that time, a supplementary selection process can be performed to identify E. BS A subset of basis states is used to reduce the selection to a subset for the subspace Hamiltonian. K Each of the following is a basis vector state. Similarly, when E... BS < K At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states in each basis state are used for the subspace Hamiltonian (e.g., K remainder = K desired - E BS Once the Hamiltonian for constructing the subspace is determined... K The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0079] Overlapping partition selection protocol with partition bias Figure 3C This is a flowchart of a method 300c for implementing an overlapping partition selection protocol with partition bias, according to an embodiment of this disclosure. The overlapping partition selection protocol is sampled from a quantum device. N Choose a set from the basis states. K each basis state ( K ≤ N ),in K Each selected basis state is in the same graph partition as the known reference state. In other words, method 300c applies an overlap criterion to select sampled basis states to be included in the subspace Hamiltonian from the complete Hamiltonian, wherein the overlap criterion is satisfied based on the following steps: generating a graph of multiple nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; identifying the reference node associated with the known reference state from the plurality of nodes; and selecting all nodes from the plurality of nodes that share a partition in the graph with the reference node.

[0080] box 310c - Sampling from quantum devices N basis vectors state At box 310c, the quantum computing system samples the experimental wavefunction to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 .

[0081] box 320c - Graph construction At box 320c, the classical computing system samples this set of data from the quantum device.N Each basis state in the 1 basis states creates nodes / vertices to construct a graph (such as...). Figure 4A (See Figure 405a). Each basis state then has a corresponding node / vertice in the graph.

[0082] box 330c - Add edge At box 330c, the classical computational system adds an edge for every combination of the two nodes / vertices representing the two basis states |b1〉 and |b2〉 in the graph, where 〈b1|H|b2〉 is nonzero. Therefore, as... Figure 4A As shown in Figure 405a, edges are constructed between some of these nodes / vertices, and some groups of nodes / vertices (e.g., graph partitions) may not be connected to other groups of nodes / vertices (e.g., other graph partitions).

[0083] box 340c - Retain nodes in the shared graph partition that are reference nodes. At box 340c, classical computational systems preserve basis states associated with known reference states in shared partitions of the graph. Two basis states are in the same partition if and only if their corresponding nodes / vertices are connected by a path. For example, in Figure 4A In this context, if node 415a is identified as a reference state, other nodes 415b-l in the same partition as node 415a are retained, while nodes 415m-o and 415p-t are removed from consideration. As used herein, the number of basis states retained in the shared partition is denoted as P. nodes .

[0084] box 350c - Supplementary options / Remove At box 350c, basis states are selected from those identified as belonging to the same graph partition as the reference state. When P nodes = K (or when P) nodes = K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when P... nodes > K At that time, a supplementary selection procedure can be performed to identify P. nodes A subset of basis states is used to reduce the selection to a subset for the subspace Hamiltonian. K Each of the following is a basis vector state. Similarly, when P... nodes < K At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states in each basis state are used for the subspace Hamiltonian (e.g., K remainder =K desired - P nodes Once the Hamiltonian for constructing the subspace is determined... K The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0085] Overlapping partition selection protocol with breadth-first bias Figure 3D This is a flowchart of a method 300d for implementing an overlapping partition selection protocol with breadth-first bias, according to an embodiment of this disclosure. Nodes with fewer “hops” (requiring fewer edges to be traversed) from the reference node are included in the subspace Hamiltonian in priority over nodes that are farther from the reference node or in different partitions. In other words, method 300d applies an overlap criterion to select sampled basis states to be included in the subspace Hamiltonian from the full Hamiltonian, wherein the overlap criterion is satisfied based on the following steps: generating a graph of a plurality of nodes from the plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; identifying the reference node associated with a known reference state from the plurality of nodes; and selecting all nodes from the plurality of nodes that are a distance of one edge from the reference node or a previously selected node, until all nodes from the plurality of nodes that share a partition with the reference node have been selected, or until at least one node from the plurality of nodes has been selected. K Each node is determined by the first occurrence.

[0086] box 310d - Sampling from quantum devices N basis states At frame 310d, the quantum computing system samples the experimental wavefunction from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 .

[0087] box 320d - Graph construction At box 320d, the classical computing system samples this set of data from the quantum device. N Each basis state in the 1 basis states creates nodes / vertices to construct a graph (such as...). Figure 4B (See Figure 405b). Each basis state then has a corresponding node / vertex in the graph.

[0088] box 330d - Add edge At box 330d, the classical computational system adds an edge for every combination of the two nodes / vertices representing the two basis states |b1〉 and |b2〉 in the graph, where 〈b1|H|b2〉 is nonzero. Therefore, as... Figure 4B As shown in Figure 405b, edges are constructed between some of these nodes / vertices, and some groups of nodes / vertices (e.g., graph partitions) may not be connected to other groups of nodes / vertices (e.g., other graph partitions).

[0089] box 340d - Breadth-first expansion At box 340d, the classical computational system identifies which basis states to preserve based on a breadth-first tree expansion starting from nodes / vertices with known reference states. Figure 4B Compared to Figure 405b, Figure 4C Figure 405c illustrates the selection of nodes 415b, 415c, and 415i. This breadth-first expansion extends from the reference state node / vertex, first including the basis states of its nodes / vertex connected to the reference state node / vertex in the graph. Next, it includes the basis states of its nodes / vertex connected to nodes / vertex connected to the reference state node / vertex in the graph, such as... Figure 4D As shown in Figure 405d (e.g., via nodes 415d, 415g, 415j, and 415l), and so on. This expansion continues until... K A node / vertices of a basis state have been included in the selection, or there are no more basis states available to be included in the graph partition containing the reference state node / vertices. As used in this paper, the number of basis states retained in the partition due to breadth-first expansion is denoted as P. nodes .

[0090] box 350d - Supplementary options At frame 350d, select the most... K Several basis states are available for selection (e.g., once identified). K The nodes of the basis vector states can be terminated according to the expansion of box 340d). When P nodes = K (or when P) nodes = K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when P... nodes < K At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states in each basis state are used for the subspace Hamiltonian (e.g., K remainder =K desired - P nodes Once the Hamiltonian for constructing the subspace is determined... K The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0091] Overlapping partition selection protocol with optimal priority bias Figure 3E This is a flowchart of a method 300e for implementing an overlapping partition selection protocol with optimal priority bias, according to an embodiment of this disclosure. In other words, method 300e applies an overlap criterion to select sampled basis states to be included in a subspace Hamiltonian from the complete Hamiltonian, wherein the overlap criterion is satisfied based on the following steps: generating a graph of a plurality of nodes from the plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; assigning a score to each of the plurality of nodes based on a heuristic measure of the importance of the corresponding basis states; identifying a reference node from the plurality of nodes that is associated with a known reference state; and selecting the next node from the plurality of nodes that has the highest score relative to all other nodes from the plurality of nodes that have ... that have the highest score relative to all other nodes that have the highest score relative to all other nodes that have the highest score relative to all other nodes that have the highest score relative to all other nodes that have the highest score relative to all other nodes that have the highest score relative to all other nodes that have the highest score relative to all other nodes that have the highest score relative to all other nodes that K Each node is determined by the first occurrence.

[0092] box 310e - Sampling from quantum devices N basis states At box 310e, the quantum computing system samples the experimental wavefunction to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j |2.

[0093] box 320e - Graph construction At box 320e, the classical computing system samples this set of data from the quantum device. N Each basis state in the 1 basis states creates nodes / vertices to construct a graph (such as...). Figure 4E (See Figure 405e). Each basis state then has a corresponding node / vertex in the graph.

[0094] box 330e - Add edge At box 330e, the classical computational system adds an edge for every combination of the two nodes / vertices representing the two basis states |b1〉 and |b2〉 in the graph, where 〈b1|H|b2〉 is nonzero. Therefore, as... Figure 4E As shown in Figure 405e, edges are constructed between some of these nodes / vertices, and some groups of nodes / vertices (e.g., graph partitions) may not be connected to other groups of nodes / vertices (e.g., other graph partitions).

[0095] box 340e - Assigning heuristic scores At box 340e, the classical computational system assigns a score to each node in the graph based on heuristic metrics such as the relevance of the corresponding basis states, the relative importance of a given basis state, etc. In various implementations, this is achieved by constructing basis states... b i Heuristic scores are calculated by overlapping with a reference node or another currently selected node.

[0096] box 350e - Best Priority Expansion At box 350e, the classical computational system identifies which basis states to retain based on a best-priority tree expansion starting from a node / vertex with a known reference state. Then, while growing the tree structure, this expansion selects the next highest-scoring node / vertex within the partition to which the reference node belongs that is allowed to be expanded. In various implementations, after the current node is expanded, the score of each unexpanded node is redistributed by the classical computational system. The tree structure grows in this way until K nodes / vertexes are expanded or all nodes within a given partition are expanded. As used herein, the number of basis states retained in a partition due to the best-priority expansion is denoted as P. nodes .

[0097] Example best-of-priority extension like Figure 4E As shown in Figure 405e, the initial selection of the basis states to be included begins with reference node 415a. The expansion proceeds by selecting the second node 415b based on its highest score among all nodes 415 linked via edges to any currently selected node. Figure 4F In Figure 405f (note that node 415h has a higher score than node 415b, but is not selected because node 415h is not connected to any currently selected node 415 via an edge). As shown in Figures 405g and 405h, the expansion can continue by selecting the highest-scoring node among the currently selected nodes that is connected via an edge.

[0098] box 360e - Supplementary options At frame 360e, select the most... K Several basis states are available for selection (e.g., once identified). K The nodes of the basis vector states can be terminated according to the expansion of box 350e). When P nodes = K (or when P) nodes = K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when P... nodes < K At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states in each basis state are used for the subspace Hamiltonian (e.g., K remainder = K desired - P nodes Once the Hamiltonian for constructing the subspace is determined... K The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0099] Contribution-based iterative selection protocol Figure 3F This is a flowchart of a method 300f for implementing a contribution-based iterative selection protocol according to an embodiment of this disclosure. In other words, method 300f applies a contribution-based iterative criterion to select sampled basis vector states to be included from the complete Hamiltonian in the subspace Hamiltonian, wherein the contribution-based iterative criterion is satisfied based on the following steps: applying a symmetry criterion to the set... N A set of sampled basis states is obtained. S 1 remaining basis vector states; based on their probabilities | α i | 2 For this group S basis states b i Sort; sort by probability | α i | 2 of K basis states b i Conduct experimental selection; use the selected... K Each basis vector constructs a subspace Hamiltonian. H K And solve H K Feature solutions; based onH K Each basis state in the characteristic solution b i amplitude α i ,from K Experimental selection of each basis vector state M A subset of basis vector states; for those from S Each basis state not included in the experimental selection b i Assign significance scores, where the significance score for each basis state is defined as follows: sig ( b i ) = | eig ( H M ) - eig ( H M+bi The option that was not included in the experimental selection and has the highest significance score is selected. R Each basis state; and the basis states of the selected combination. M and R .

[0100] box 310f - Sampling from quantum devices N basis states At box 310f, the quantum computing system samples the experimental wavefunction to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b i The probability of being sampled is | α i | 2 .

[0101] box 320f - Remove basis states according to the symmetry criterion and then sort the basis states. At box 320f, the classical computational system removes basis states that do not satisfy the symmetry criteria from the consideration of the subspace Hamiltonian, where the symmetry criteria include: removing basis states that do not have a bit string containing a number of "1"s that matches the number of electrons in the chemical system being analyzed, or removing basis states that do not have a bit string containing a number of "1"s at positions corresponding to a given type of orbital that equals the number of electrons in that given type of orbital of the chemical system represented by the full Hamiltonian. As used herein, the number of basis states that satisfy the symmetry criteria is referred to as S Then, the classical computer, based on each b i probability | α i | 2 Come to S basis states inb i Sort them.

[0102] box 330f - K Experimental selection of basis vector states At box 330f, the classical computing system... S Among the remaining basis states, the pair with the highest probability | α i | 2 of K Experimental selection of each basis vector state was carried out.

[0103] box 340f - Calculate the characteristic solution of the Hamiltonian of the experimental subspace At box 340f, the classical computing system utilizes from S experimental selection K Using basis vectors to construct the subspace Hamiltonian H K Furthermore, we solve for the characteristic solution of the Hamiltonian of this subspace.

[0104] box 350f - from K Experimental selection of each basis vector state M basis states At box 350f, the classical computational system is based on each basis state in the eigensol. b i amplitude α i from K Experimental selection of each basis vector state M basis states b i subset of (where) M < K Each amplitude α i Indicator basis states b i The degree of contribution to the characteristic solution. Then, the classical computing system utilizes... M The chosen basis states are used to construct the subspace Hamiltonian. H M .

[0105] box 360f - Construct significance scores for basis states that are not in experimental selection At frame 360f, the classic computer... S Not included in frame 330f K Each basis state in the experimental selection of basis states b i Assign significance scores to each basis state. b i The significance score gives b i subspace Hamiltonian H KThe significance score is an indicator of the importance of the characteristic solutions. In various implementations, this significance score (referred to herein as...) sig ( b i The formula for calculating () is: sig ( b i ) = | eig ( H M ) - eig ( H M+bi This fraction will be the Hamiltonian of the subspace. H M Characteristic solutions and the use of basis vectors M + b i The characteristic solutions of the constructed subspace Hamiltonian are compared, which indicates that... b i This includes the effects produced by the selection of basis states.

[0106] box 365f - Use significance scores to select R basis states At box 365f, the classical computing system originates from this group. S Select from the basis states that have the preceding... R A group with the highest significance score R Each of the basis vector states. This choice represents the improvement of the subspace Hamiltonian. H M The set of characteristic solutions with the highest observational influence R basis states b i .

[0107] box 370f - Will M The experimental basis vectors and R A combination of basis vector states At box 370f, the classical computational system will find the eigenvalue with the largest amplitude. α i A group M basis states b i With the highest significance score sig ( b i A group of ) R Combining basis vector states. As used in this paper, including in the group M+R The number of basis states in the vector is called E BS Therefore, the classical computational system generates a set of (up to) K basis states, which have been evaluated as having significant meaning for representative characteristic solutions of the generative chemical system.

[0108] box 380f - Supplementary options / Remove At box 380f, from E, which is identified as satisfying the electron conservation choice BS Select the most from the basis vectors K Several basis states are available for selection. When E BS = K (or when E) BS = K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when E... BS > K At that time, a supplementary selection process can be performed to identify E. BS A subset of the basis vector states is selected and removed from the selection to reduce the number of Hamiltonians used in the subspace. K Each of the following is a basis vector state. Similarly, when E... BS < K At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states in each basis state are used for the subspace Hamiltonian (e.g., K remainder = K desired - E BS Once the Hamiltonian for constructing the subspace is determined... K The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0109] α / β Exchange Supplemental Agreement Figure 3G This is a flowchart of a method 300g for implementing an α / β exchange supplementation protocol according to an embodiment of this disclosure. In method 300g, additional basis states from the symmetry space are used to supplement basis states selected via the application of a single selection protocol or a series of selection protocols. These additional states are constructed by separating the selected basis states into their α and β configurations, and then performing a permutation to exchange the α and β configurations, thereby creating new basis states. If a basis state with a certain α and β configuration is found to be important to the eigenvalue, it is usually found that a basis state with the corresponding exchanged α and β configuration is also important to the eigenvalue.

[0110] box 310g - Sampling from quantum devices N basis states At frame 310g, the quantum computing system samples the experimental wavefunction to obtain a value from 2... n A set of N basis states is obtained from the given basis states, where each basis state b j The probability of being sampled is |α j | 2 .

[0111] box 320g - Application of selection methods At box 320g, classical computing systems apply a single selection method or a series of selection methods to identify samples taken from quantum computing systems. N A subset E of the basis states that satisfies this selection protocol init This state selection forms the core space, and the additional states selected by method 300g will be attached to this core space.

[0112] box 330g - Replacement α and β Configuration to construct additional basis states At frame 330g, the classical computing system will use E init Each basis state in b i Separately into its α and β configurations, i.e. α i and β i Then, the classical computing system performs a permutation that swaps the α and β configurations to obtain E. init Each basis state in b i Obtain new basis states b' i This forms a system containing basis vector states. b' i Group E supp .

[0113] box 340g - Add the constructed basis states to the basis states of the selected group. At frame 340g, the classical computing system selects the basis vector state E. init and E supp The choice E for combining to form a unique basis state BS By utilizing E supp Supplementing core space E init The core space is expanded to include important basis states that were not obtained through sampling by the quantum computing system.

[0114] box 350g - Perform (optional) supplementary selection / Remove At frame 350g, the most [value] basis states are selected from those identified through the initial selection and further supplemented by the α / β exchange method. K Several basis states are available for selection. When E BS = K (or when E) BS = K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when E... BS > K At that time, a supplementary selection process can be performed to identify E. BSA subset of basis states is used to reduce the selection to a subset for the subspace Hamiltonian. K Each of the following is a basis vector state. Similarly, when E... BS < K At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states in each basis state are used for the subspace Hamiltonian (e.g., K remainder = K desired - E BS Once the Hamiltonian for constructing the subspace is determined... K The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0115] Heuristic supplementary protocol In Method 300h, a heuristic is used to guide the selection of additional basis states from the symmetric space to be included in the core space. The heuristic for each basis state is the Hamming distance of that basis state relative to the Hartree-Fock basis states. It is well known that Hartree-Fock basis states contribute significantly to eigenvalues, and it has been observed that the importance of a basis state to eigenvalues ​​is generally inversely proportional to the degree of difference between the basis state in question and the Hartree-Fock basis states.

[0116] box 310h - Sampling from quantum devices N basis states At frame 310h, the quantum computing system samples the experimental wavefunction to obtain a value from 2... n A set of N basis states is obtained from the given basis states, where each basis state b j The probability of being sampled is |α j | 2 .

[0117] box 320h - Select the application of the protocol At box 320h, classical computing systems apply a single selection protocol or a series of selection protocols to identify samples taken from quantum computing systems. N A subset E of the basis states that satisfies this selection protocol init This state selection forms the core space, and the additional states selected by method 300h will be attached to this core space.

[0118] box 330h - Select a random subset of effective basis states At frame 330h, the classical computational system belongs to a symmetric space but is not included in E. init Selecting a random subset E from the basis vector states rand Choose E randThis forms the domain for probability sampling in subsequent operations of method 300h.

[0119] box 340h - Calculate the Hamming distance for each basis state. At frame 340h, the classical computing system for E rand Each basis state in b i Calculate the Hamming distance relative to the Hartree-Fock basis states. h i Basis state b i Hamming distance h i yes b i The number of positions that differ from the Hartree-Fock basis states.

[0120] box 350h - Construct the probability distribution of the sample At frame 350h, the classical computing system is E. rand Each basis state in b i Assign sampling probability ,in Then from E rand A certain number of basis states are sampled to supplement the selection of E. init And define the sampling probability. p i This makes the basis vector state b i The probability of sampling is proportional to its similarity to the Hartree-Fock basis states.

[0121] box 360h - Sampling of basis states from the selected subset At frame 360h, the classical computing system is based on... p i The probability distribution formed is used to perform operations on E rand The basis states are sampled to form a set of sampled basis states E. supp It complements the core space E formed in frame 320h. init .

[0122] box 370h - Add the sampled basis states to the basis states of the selected group. At frame 370h, the classical computational system selects the basis state E. init and E supp The choice E for combining to form a unique basis state BS By utilizing E supp Supplementing core space E init The core space is expanded to include important basis states that were not obtained through sampling by the quantum computing system.

[0123] box 380h - Perform (optional) supplementary selection At frame 380h, select from the basis states identified through the initial selection and further supplemented by a heuristic method. K Several basis states are available for selection. When E BS = K (or when E) BS = K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when E... BS > K At that time, a supplementary selection process can be performed to identify E. BS A subset of basis states is used to reduce the selection to a subset for the subspace Hamiltonian. K Each of the following is a basis vector state. Similarly, when E... BS < K At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states in each basis state are used for the subspace Hamiltonian (e.g., K remainder = K desired - E BS Once the Hamiltonian for constructing the subspace is determined... K The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0124] fixed K Maximum value selection protocol Figure 3I It is used to achieve fixed K The flowchart of the maximum value selection protocol method 300i is shown. This method has a fixed... K Values, of which are sampled from the quantum device N Select the basis states with the highest (or highest absolute) value. K Each basis vector state. In various implementations, although it is possible to increase or decrease the number of basis vector states across iterations. K The magnitude of the value has advantages, but when a known number of basis states is required, having a known fixed value is not ideal. K Values ​​provide a simple solution. For example, if one or more selection protocols described in this article are executed and this results in fewer than [a certain number of] selections being made... K desired After obtaining the basis vector states, the system determines that a supplementary selection should be performed, and this can be applied to fixed basis vector states. K Value (e.g., K) fixed ,in Kfixed = K desired - K selected Execute method 300i to satisfy the condition to be added from one or more previous iterations. K selected The number of basis states in each basis state, thus satisfying the requirement of K desired The number of basis states selected. Similarly, if executing one or more selection protocols described in this paper results in more than [number of basis states selected] being chosen... K desired After obtaining the basis states, the system determines that supplementary removal should be performed, and a fixed method can be used. K The value is used to execute method 300i to retain only the value from the initial selection. K desired Each basis vector state.

[0125] box 310i - Sampling from quantum devices N basis states At box 310i, the quantum computing system samples the experimental wavefunction to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 .

[0126] box 320i - Ranking of basis states based on probability At box 320i, for the basis vector states sampled according to box 310i b j Classical computing systems are based on their probability values ​​| α j | 2 Sort them from highest to lowest. In various implementations, a variety of sorting algorithms can be used (e.g., bubble sort, heap sort, merge sort, tree sort, insertion sort, shell sort, etc.).

[0127] box 330i - choose K The highest value of the basis state At box 330i, the classical computing system selection is sorted according to having the previous K The highest value K Each basis vector state.

[0128] fixed K Example of the concept of the maximum value selection protocol Figure 4I An example is given of selecting from sorted basis states according to method 300i. K Example cutoff thresholds for each basis state. In the illustrated example,K The cutoff threshold of 425 is fixed at four (e.g., K =4), such that the four basis states with the highest sampling probability are selected, and other basis states are excluded / omitted. Several basis states 435a-j are illustrated as being sorted from highest to lowest probability, where four basis states 435a-d are on the included side of the truncation threshold 425, and six basis states 435e-j are on the excluded side of the truncation threshold 425. As will be understood, the labels of basis states 435a-j are given for the convenience of the reader and correspond to the probability values ​​of these basis states after sorting, which may be independent of the position of the basis state in the Hamiltonian matrix representation.

[0129] Use fixed K Notes on the Maximum Value Selection Protocol As will be understood, while the fixed K maximum selection protocol facilitates the selection of a predefined number of basis states, it focuses on a single selection criterion: sampling probability. Therefore, the third basis state 435c may be preferentially selected for inclusion over the fifth basis state 435e, although the fifth basis state 435e has a greater impact on computational accuracy than the third basis state 435c, depending on one or more other selection criteria (e.g., those discussed with regard to methods 300a-f). Similarly, the fixed K maximum selection protocol may also result in basis states with equal values ​​being split between included or excluded groups (e.g., basis states 435d and 435e). Furthermore, sorting algorithms are generally considered computationally intensive, and for large numbers of basis states (e.g., 2^35c ... n Sort the data to generate an ordered list to which a truncation threshold of 425 can be applied, which can consume a lot of time and resources.

[0130] threshold - Probability Selection Protocol Figure 3J This is a flowchart of method 300j for implementing a threshold-probability selection protocol, where samples are taken from a self-quantum device. N The basis states are selected from a given threshold value. In various implementations, the number of selected basis states may be much greater or much less than... K desired Furthermore, various thresholds can be checked until the number of selected basis states is within a certain range. K desired Within the expected range (e.g., K desired ±10%). Beneficially, the threshold-probability selection protocol, despite being in the unknown... K selectedIt operates under certain conditions, but is computationally easy to implement (e.g., avoiding the need to perform computationally complex sorting algorithms; thus allowing operations on unsorted lists or pools of basis states), and allows for rapid repetition using a variety of different probability thresholds. Therefore, the threshold-probability selection protocol can be used as an initial selection protocol (determined by supplementing or omitting other selection protocols), or as a supplement to or omitting of another selection protocol with low computational overhead.

[0131] box 310j - Sampling from quantum devices N basis states At box 310j, the quantum computing system samples the experimental wavefunction to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 .

[0132] box 320j - Ranking of basis states based on probability At box 320j, the classical computational system selects all those basis states that have a probability value above a threshold, sampled according to box 310j.

[0133] box 330j - Sure K selected Is it close? K Desired At box 330j, the classical computational system determines the number of basis vector states selected according to box 320j. K selected Is it close to the number of basis states to be selected? K Desired When K selected In K desired When within the predefined window, method 300j can proceed to box 340j. When K selected Not in K desired When within the predefined window, method 300j can return to box 320j to select all basis states above different thresholds, or it can proceed to another selection protocol.

[0134] box 340j - Supplementary options / Remove At box 340j, once K selected The basis states are close to K desired Or, if the threshold selection protocol is deemed unsuitable for computing feature solutions. K desired After that, we will proceed with this. K selected Each fundamental vector state is supplemented to achieve... K desired .when Kselected = K desired (or when) K selected = K remainder That is, from an earlier selection process K The selection process can end when the unfilled portion is selected. However, when... K selected > K desired At that time, a supplementary selection process can be performed to identify. K selected A subset of basis vector states, to be used for the initial selection of the subspace Hamiltonian. K selected Remove from each of the basis vector states, thereby removing K selected Reduce to K desired Similarly, when K selected < K desired At this time, a supplementary selection process can be performed to identify which items should be included. K Additional basis states from the existing basis states are used to construct the subspace Hamiltonian. Once the basis states for constructing the subspace Hamiltonian are determined... K The system can then use these basis states (using one or more selection protocols). K Calculate and output the characteristic solutions for each basis vector state.

[0135] threshold - Example of the concept of probabilistic selection protocol Figure 4J An example is given for selecting from unsorted basis states according to method 300j. K An example of the threshold T for each basis vector state. In the illustrated example, two thresholds 445a and 445b are shown, where the first threshold 445a is greater than the second threshold 445b (e.g., x > y). For ease of understanding and in relation to... Figure 4I In comparison, Figure 4I and Figure 4J The basis state 435a-j (and its probability) given in the text are exactly the same. For example... Figure 4J As illustrated, when T=x for the first threshold 445a, three basis states 435a-c are selected; and when T=y for the second threshold 445b, five basis states 435a-e are selected. As will be understood, basis state 435a-j in... Figure 4J The order in Figure 4I This differs from the previous approach, allowing computer systems to omit the execution of sorting algorithms used to select basis states.

[0136] Threshold selection In various implementations, the threshold value can be based on a user-defined value or a value derived from the sampled basis states, such as percentile probability values. In some implementations, basis states with outliers or those a priori identified as to be included (e.g., from an earlier selection process) can be excluded from the calculation when deriving the threshold. For example, Figure 4J In x This might correspond to the 60th percentile, making the basis states with the highest probability values ​​among the top 40%. Figure 4J In this context, y might correspond to the 50th percentile, allowing the selection of the half of the basis states with the highest probability values. As will be understood, a percentile-based threshold may not always select basis states whose number equals a percentage of the total number of selected basis states. Continuing with the example where x is set to the 60th percentile... Figure 4J Only 30% of the total basis states (e.g., three of the ten basis states 435a-j exemplified) are selected because although x is mathematically equal to the 60th percentile, there is no probability of x for four basis states (e.g., 40% of the ten basis states 435a-j exemplified).

[0137] Additional selection process As discussed regarding electron conservation selection processes (e.g., symmetry criteria), overlap partitioning selection processes (e.g., overlap criteria), threshold-based selection processes, etc., when a selection process results in more or fewer basis states... K At this time, various supplementary selections can be made. In various implementations, the operator can rerun the selection process using different selection criteria, or supplement the first selection criterion with a second selection criterion. As a supplement or alternative to the electron conservation and overlap partitioning selection process, the operator can randomly select to include (or exclude) by manual selection or by setting a threshold for the amplitude of the basis states (e.g., a screening criterion for amplitude values ​​higher than a given threshold). K The basis states are selected from the basis states. Alternatively, the operator may use values ​​selected from previous iterations of the selection process (or different sampled values ​​from the test wavefunction, or different preparations of the test wavefunction).

[0138] Benefits of Choice By not only choosing how many basis states from the complete Hamiltonian to include in the subspace Hamiltonian (e.g.) K Furthermore, by selecting, via one or more selection protocols, which basis states of the full Hamiltonian are included in the subspace Hamiltonian, the operator can alter the representativeness of the subspace Hamiltonian to the full Hamiltonian. Therefore, by being able to change the size of the subspace Hamiltonian (via adjusting...),K Both the choice of the basis vectors used to construct the subspace Hamiltonian and the selection of different selection protocols allow operators to easily explore and test different settings to allow for more efficient or accurate computation of eigenvalues. These choices (regardless of the specific context) K Each choice in the selection protocol can be performed independently of other choices via an iterative process, such as regarding relative to... Figure 5A As described in method 500a.

[0139] According to the diagram 5A Methods for determining ground state value and ground state energy Figure 5A This is a flowchart of an example method 500a for determining the ground-state value and ground-state energy of a Hamiltonian, according to an embodiment of this disclosure, which can be used to represent a chemical system. Method 500a begins at block 505a, where the complete Hamiltonian representing the chemical system is created. Method 500a then continues with one or more iterations of blocks 510a-565a to solve for a subspace Hamiltonian of the chemical system using both quantum and classical computing resources, which represents the complete Hamiltonian in a classically tractable problem space. Because classical computing resources are more readily available than quantum computing resources, various techniques can be used to attempt handover multiple times in different iterations to make the computation tractable for classical computers, while still providing an improvement in overall system efficiency compared to determining the ground-state value and ground-state energy of the complete Hamiltonian entirely via quantum computing.

[0140] method 500a A simplified understanding Figure 5B Provided information about Figure 5A A simplified or overview understanding of Method 500a is provided to aid in deeper understanding and highlight the improvements offered by Method 500a compared to conventional procedures for determining the Hamiltonian ground state value and ground state energy. In various embodiments, the high-level heuristics identified in box 530b may include one or more selection protocols discussed herein (e.g., as relative to...). Figures 3A to 3J The state of the quantum circuit is updated by feedback loops from boxes 550b, 560b, 570b and 510b (when the eigenvalues ​​have not yet converged) to perform subsequent analysis iterations using the updated wavefunction based on the previous analysis.

[0141] method 500b Description Usually, such as Figure 5BAs shown in method 500b, a sampled state (|S〉= ∪ (θ)|ψ0〉) is prepared on a quantum device (according to box 510b), and Ns samplings of the quantum state are performed in the computational basis (according to box 520b). This is based on various high-level heuristics and selection protocols, such as those concerning... Figures 3A to 3J The ones discussed (according to box 530b) select K basis states from Ns sampled basis states, where K ≤ Ns. The classical computational system constructs the subspace Hamiltonian H based on the selected K basis states. K Furthermore, at the classical level, the subspace Hamiltonian H... K Diagonalization is performed to obtain the target eigenvalue c and eigenstate c. K (According to box 540b). When the value of the target feature value c converges (according to box 550b), method 500b can terminate; otherwise, method 500b proceeds to box 560b, where the latest value of the target feature value c is used to update the state θ. Then, the classical computer updates the initial state such that |ψ0〉=|ψK〉 (according to box 570b), and method 500b returns to box 510b to prepare a new sampled state based on the new initial state (from box 570b).

[0142] method 500a , 600 and 700 Relationship about Figure 6 Method 600 described provides alternatives and supplementary understandings to Method 500a when using Pauli sampling or Pauli measurement to analyze Hamiltonian quantities, while regarding Figure 7 The described method 700 provides an alternative and supplementary understanding to method 500a when analyzing Hamiltonians without using Pauli sampling or Pauli measurements. These experimental wavefunctions and other functional equivalents can be analyzed as a single parameterized quantum circuit (also known as a hypothetical) or multiple / ensemble parameterized quantum circuits.

[0143] box 520a With Method 600 Relationship In some implementations, the wavefunction protocol for generating the test wavefunction includes a third and multiple iterations via a hybrid quantum-classical computing system running a variational quantum characteristic solver (VQE) until a third stopping criterion is met. Each iteration of this third and multiple iterations includes: constructing a parameterized quantum circuit based on an initial state via a classical computing system; generating a parameterized wavefunction approximating the ground state of the complete Hamiltonian via executing the parameterized quantum circuit on the quantum computing system; performing Pauli sampling on the parameterized wavefunction generated by the parameterized quantum circuit with respect to the complete Hamiltonian via the quantum computing system; calculating the expectation value of the parameterized wavefunction relative to the complete Hamiltonian via a classical computing system; and updating a set of parameters of the parameterized quantum circuit based on the expectation value; stopping the third and multiple iterations on the hybrid quantum-classical computing system in response to the satisfaction of the third stopping criterion, wherein the parameterized wavefunction from the last iteration of the third and multiple iterations is output; and wherein the third and multiple iterations are performed before the second and multiple iterations. These operations will be related to... Figure 6 Let's discuss this in more detail.

[0144] box 520a With Method 700 Relationship In some implementations, the experimental wavefunction preparation protocol includes: constructing a parameterized quantum circuit based on an initial state via a classical computing system; generating a parameterized wavefunction of an approximately complete Hamiltonian ground state by executing the parameterized quantum circuit on a quantum computing system, without performing Pauli sampling; and outputting the parameterized wavefunction. These operations will relate to... Figure 7 Let's discuss this in more detail.

[0145] By hybrid quantum - Classical computing system execution Figure 5A The operations are performed by a hybrid quantum-classical computing system. The classical computing system typically performs the operations shown to the left of the dashed line, while the quantum computing system typically performs the operations shown to the right of the dashed line. As will be understood, some operations may be performed by both the classical and quantum computing systems, or involve the transfer of data between the classical and quantum computing systems; these operations are omitted for ease of understanding. For example, preparing an experimental wavefunction (according to box 515a) involves operations performed by both the classical and quantum computing systems, but is typically illustrated as being performed by the quantum computing system.

[0146] box 505 - Create a complete Hamiltonian At box 505a, a classical computational system is used to create a complete Hamiltonian representing the chemical system. The complete Hamiltonian has a value with 2... nThe Hilbert space of 2 basis states represents all possible states that the quantum system can exist in. This contrasts with the subspace Hamiltonian (discussed in detail in box 540a), which includes 2 basis states chosen to best represent the desired properties of the chemical system. n A subset of basis vector states. The Hamiltonian is used to evaluate the energy of the states that a chemical system may occupy, while the subspace Hamiltonian provides a computationally simpler platform to evaluate that energy state, which can be evaluated by classical computing systems. In contrast, the full Hamiltonian may be too complex for classical computing systems to handle within a reasonable time frame, thus requiring the use of quantum computing systems for evaluation.

[0147] box 510a - Assign initial state At box 510a, the classical computing system assigns an initial state to the complete Hamiltonian for the quantum computing system to begin iterative analysis. The initial state represents the complete set of 2... n The initial allocation of amplitudes of the basis states is the starting point for computation. In various embodiments, classical computing systems can use various state preparation protocols to assign initial states to the complete Hamiltonian. These initial states can include at least one of the following: states prepared via the Hartree-Fock protocol, zero states, computational basis states in Hilbert space, states prepared via ab-initialization initial state preparation protocols (e.g., density functional theory (DFT), configuration interactions (CI), coupled clusters (CC), Møller-Plesset perturbation theory (MPn), etc.), states prepared via tensor network initial state preparation protocols (e.g., optimizing a tensor network and mapping the resulting states to quantum circuits to obtain quantum states), sparse initial states from the final iterative characteristic solutions of a previous set of iterative analyses from a hybrid computing system (e.g., the final iteration of previous iterations for subsequent iterations), uniformly distributed states, and randomly distributed states. This disclosure envisions that different protocols might be used when subsequent iterations are performed on the same chemical system by a quantum computing system.

[0148] From the box 510a To the box 510a The first loop iteration Different computations are performed using classical and quantum computing systems, and an evaluation iteration (and adjustments made thereto) based on the initial state is executed on a hybrid quantum-classical computing system. This evaluation is performed by running the first multiple iterations (e.g., as a loop) on the hybrid quantum-classical computing system until a first stopping criterion is met (as discussed with respect to box 555a), and may include various sub-loops for different analyses.

[0149] box 515a - Generate experimental wave function At box 515a, the quantum computing system generates, based on known quantum mechanical practices, a trial wave function (||) describing the quantum state of an isolated quantum system. Ψ The Hamiltonian is analyzed through multiple iterations starting from the initial state to find an approximation of the characteristic solution of the Hamiltonian.

[0150] box 520a - Sampling test wave function At box 520a, the quantum computing system samples the test wavefunction generated according to box 515a in the computational basis to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 In conjunction with the information about Figure 6 Method 600 or about described Figure 7 The described method 700 provides a more detailed understanding of the computational basis. In various implementations, sampling the test wavefunction updates the values ​​of various basis states based on the initial basis states (according to box 510a) or earlier iterations of sampling the test wavefunction.

[0151] box 525a - Second cycle At block 525a, the classical computing system determines whether a second stopping criterion has been met. In various implementations, the second stopping criterion is met in response to at least one of the following: a predefined number of iterations of the second loop from block 520a to block 525a have been performed; and the batch results contain at least a predefined number of values. When the second stopping criterion has not been met, method 500a returns from block 525a to block 520a for the quantum computer to continue performing additional iterations of the second loop. When the second stopping criterion has been met, method 500a proceeds from block 525a to block 530a, thereby exiting the second loop.

[0152] box 530a - Post-processing At box 530a, the classical computational system performs post-processing on the basis states to obtain results that satisfy standards including [the relevant criteria]. K Each basis state is post-processed to evaluate each basis state according to a criterion to determine whether it should be used to construct a subspace Hamiltonian from the full Hamiltonian. In various implementations, the criterion is satisfied in response to at least one of the following: the second plurality of values ​​contains a predefined number of values ​​from the first plurality of values; the number of values ​​contained in the second plurality of values ​​is within a predefined threshold of the predefined number of values ​​from the first plurality of values; or the second plurality of values ​​satisfies one or more selection protocols, such as regarding... Figures 3A to 3J As described.

[0153] In the box 530a Post-processing options used In various implementations, the post-processing protocol includes a selection protocol comprising: selecting, via a classical computing system, a second plurality of values ​​from a first plurality of values ​​that satisfy at least one of the following: a symmetry criterion (e.g., checking whether the number of electrons is conserved), a screening criterion (e.g., a value above a given threshold), an overlap criterion (e.g., identifying overlap with other basis states in a basis state diagram, and identifying important groups of basis states); or randomized selection. When using a symmetry criterion, the symmetry criterion may be satisfied in response to the value representing a Hilbert space-calculated basis state with a predetermined number of electrons, or the screening criterion may be satisfied in response to the value representing a calculated basis state in Hilbert space and the magnitude value being within a predefined highest percentage of a plurality of magnitude values ​​among the first plurality of values. For example, the classical computing system may select a magnitude value that precedes the magnitude value of the basis state. X % of those basis states. Various selection protocols will be about Figures 3A to 3J Let's discuss this in more detail.

[0154] box 535a - K sufficiency of values At box 535a, the classical computing system determines whether the number of values ​​in the second plurality of values ​​is sufficient to generate the subspace Hamiltonian. The operator may specify a sufficiency threshold for the number of values ​​included in the second plurality of values ​​based on factors such as the capabilities of the classical computing system (e.g., a smaller number of values ​​results in a smaller subspace to be analyzed, leading to faster computation), the level of accuracy relative to the full Hamiltonian (e.g., a larger number of values ​​results in a larger subspace to be analyzed, which may be more representative than (or at least no less representative than) a smaller subspace), and other operator considerations. In various implementations, the classical computer can (in determining...) K Whether before or after (sufficiency) the selected basis states are supplemented with additional basis states. K The basis vector states are extended by a second plurality of values ​​via a classical computational system, by a third plurality of values ​​including at least one of the following: a second plurality of values ​​from a previously run second plurality of iterations or an external source (e.g., a library with historical or standardized data of a chemical system).

[0155] The benefits of choosing a protocol and the ease of iteration Compared to traditional computational methods, this approach provides better control over the size and composition of the Hamiltonian in the subspace via a selection protocol, and can traverse different... K Values ​​and the selection of fill KBy processing the basis vectors of each state, hybrid computing systems can save overall computational resources or alter the allocation of computational resources between classical and quantum computing systems. Furthermore, by providing multiple nodes to the hybrid system during the VQE handover process to determine when (and which elements) change in subsequent iterations, hybrid computing systems can further enhance the ability of classical computing systems to generate accurate characteristic solutions compared to traditional computational methods.

[0156] box 535a Based on K Value decision when K When the value is insufficient, too high, or too low compared to the operator-defined threshold or window, method 500a returns from box 535a to box 520a for the quantum computer to continue execution. K When the value is sufficient (e.g., falls within an operator-defined threshold or window), method 500a proceeds from box 535a to box 540a.

[0157] box 540a - Subspace Hamiltonian Construction At box 540a, the classical computational system constructs a subspace Hamiltonian to represent the chemical system based on a second plurality of values ​​selected from box 530a, including a standard. In various embodiments, the subspace Hamiltonian representation includes at least one of the following: K×K Matrices, Pauli sums, and black-box representations with element-wise access protocols.

[0158] box 545a - Calculation of minimum eigenvalue and eigenvector At block 545a, the hybrid quantum-classical computing system computes the characteristic solution of the subspace Hamiltonian. In various embodiments, the classical computing system uses a characteristic solver to compute the characteristic solution, which can be at least one of the following: a quantum computing characteristic solver that performs the computation on a quantum computing system, a classical computing characteristic solver that performs the computation on a classical computing system, or a hybrid computing characteristic solver that performs the computation on a hybrid computing system.

[0159] box 550a - Ground state value calculation At box 550a, the classical computational system uses the characteristic solutions of the subspace Hamiltonian to compute the approximate ground state value representing the ground state of the complete Hamiltonian, and the approximate ground state energy value representing the ground state energy of the complete Hamiltonian.

[0160] box 555a - The first iteration terminates. At block 555a, the classical computing system determines whether a first stopping criterion has been met, thus completing the computation loop. In various implementations, the first stopping criterion is met in response to at least one of the following: a predefined number of iterations in a first plurality of iterations has been performed; the first plurality of iterations have been run for a predefined time; a predefined time has been reached using a quantum computing system; the change in the ground state energy value from a given iteration in the first plurality of iterations to subsequent iterations in the first plurality of iterations is within a threshold of the quantum noise change value of the quantum computing system; the first derivative of the ground state energy value from a given iteration in the first plurality of iterations falls below a termination threshold; and the change in the ground state energy value from multiple previous iterations in the first plurality of iterations to subsequent iterations in the first plurality of iterations is within a termination threshold.

[0161] box 555a Decision based on the first stopping criterion If the first stopping criterion has not yet been met, method 500a proceeds from block 555a to block 565a to determine how to continue based on various decision criteria. If the first stopping criterion has been met, method 500a proceeds from block 555a to block 560a, and optionally may perform relative to... Figure 5C Overview of the operations.

[0162] box 560a - Output At box 560a, in response to the first stopping criterion being met and the classical computer ceasing the first multiple iterations, the classical computing system outputs the characteristic solution of the final iteration of the first multiple iterations. In various implementations, once the characteristic solution is known, the operator can use it via the classical computing system to simulate a state relative to the target chemical system that can be represented as 2 n ×2 n The matrix corresponds to the complete Hamiltonian of the chemical system, and the most stable configuration and properties of the system are determined. Characteristic solutions of the chemical system provide information about its reactivity, stability, and potential interactions with other chemical systems. By evaluating the reactivity, stability, and interactions of the chemical system, the operator can use it to synthesize other chemical systems or study various reactions.

[0163] box 565a - Iterative determination At box 565a, in response to the first stopping criterion not being met, the classical computing system determines which decision criteria have been met to determine how to continue. In various implementations, when the first stopping criterion has not been met, the operator is prompted to meet which of the three decision criteria to restart the computation fully or partially. In various implementations, method 500a can iterate multiple times from box 510a through box 565a until the first stopping criterion is met or the operator terminates method 500a. In various implementations, each time method 500a iterates through box 565a, at least one parameter in the computation is changed from the previous iteration to the next. For example, it could be... K Different thresholds can be selected, or different procedures can be chosen to determine the criteria. In various implementations, the classical computing system automatically determines which decision criteria to meet based on which loop is more likely to be repeated than others; while in other implementations, the classical computing system prompts the operator (or has been instructed in advance by the operator) which decision criteria should be met.

[0164] In the box 565a The first decision criterion is met. When the first decision criterion has been met, method 500a returns from box 565a to box 510a to generate a new (e.g., second) initial state, and then re-runs the first multiple iterations on the hybrid quantum-classical computing system until the first stopping criterion is met in a subsequent (e.g., second) iteration.

[0165] In the box 565a The second decision criterion is met. When the second decision criterion is met, method 500a returns from box 565a to box 520a, back to the node sampled from the experimental wavefunction, and then re-runs the first multiple iterations on the quantum computing system until the second stopping criterion is met in the subsequent iteration (e.g., the second time), and continues through boxes 525a-560a until the first stopping criterion is met in the subsequent iteration (e.g., the second time).

[0166] In the box 565a The third decision-making standard is met. When the third decision criterion is met, method 500a returns from block 565a to block 530a, whereby a classical computing system reselects values ​​for constructing the subspace Hamiltonian. In various embodiments, the classical computing system uses the previously generated first plurality of values ​​and selects new second plurality of values, each of which satisfies the inclusion criterion. In various embodiments, the classical computing system may use different inclusion criteria or different processing protocols to select the new second plurality of values. Method 500a can then continue through blocks 535a-560a until the first stopping criterion is subsequently met (e.g., a second time).

[0167] According to the diagram 5C Method 500c Operations to perform union analysis Figure 5C This is a flowchart of an optional sub-method 500c that can be executed between blocks 555a and 560a of method 500a. At block 510c, the classical computing system collects data from previous iterations of method 500a. N Some or all of the previously sampled basis states are taken, and at block 520c, a union of these previously sampled basis states is created, which are shared between iterations. In some implementations, the union is formed by previously selected basis states from previous iterations. K It consists of 10 basis states, but it can also consist of unselected basis states (e.g., NK ) or a complete set N The union is composed of sampled basis states. For example, if a first set of K1 basis states {A, B, C, D} is selected in the first iteration, and a second set of K2 basis states {D, E, F, G} is selected in the second iteration, then the union after the second iteration could be {A, B, C, D, E, F, G}. Similarly, if N1 basis states are sampled in the first iteration and N2 basis states are sampled in the second iteration, the union state allows exploration of different portions of the minimum optimal space represented in N1 and N2, respectively, within a single merged / union set of values.

[0168] box 530c - Choice within the union At box 530c, classical computing systems apply a selection protocol to the union to obtain the result from the union. U k Select K union Several basis vector states are provided for analysis. In various implementations, [the following is used for selection]... K union The selection protocol for each basis state can be the same as or different from any selection protocol used in block 530a of method 500a.

[0169] box 540c - Construction of Union Hamiltonian At box 540c, the classical computing system, according to the standard, extracts from the union. U k The subspace Hamiltonian is constructed for this union. H' Uk To represent a chemical system. In various embodiments, the subspace Hamiltonian representation includes at least one of the following: K×K Matrices, Pauli sums, and black-box representations with element-wise access protocols.

[0170] box 550c - Calculation of Hamiltonian characteristic solutions in union subspaces At box 550c, the classical computing system computes the Hamiltonian of the union subspace. H' UkThe characteristic solution. In various implementations, the classical computing system uses a characteristic solver to compute the characteristic solution, which can be at least one of the following: a quantum computing characteristic solver that performs computation on a quantum computing system, a classical computing characteristic solver that performs computation on a classical computing system, or a hybrid computing characteristic solver that performs computation on a hybrid computing system.

[0171] box 560c - Calculation of the characteristic solution of the complete Hamiltonian At box 560c, the classical computational system uses the Hamiltonian from the union subspace. H' Uk The characteristic solutions of the complete Hamiltonian are calculated from the characteristic solutions. After calculating the characteristic solutions according to boxes 550c and 560c, method 500c can proceed to box 560a of method 500a to output these characteristic solutions to the user for review.

[0172] According to the diagram 6 Methods for determining ground state value and ground state energy Figure 6 This is a flowchart of an example method 600 for improving the computational efficiency of a hybrid quantum-classical computing system when using Pauli sampling or Pauli measurement to determine the ground state value and ground state energy of a Hamiltonian, according to an embodiment of the present disclosure, wherein the Hamiltonian can be used to represent an element or compound.

[0173] As a method 500 Special cases of methods 600 about Figure 6 Method 600 of the discussion can be understood as being about Figure 5A The discussion focuses on a special case of method 500a, where Pauli sampling or Pauli measurement is performed relative to parameterized quantum circuitry. Accordingly, the operation of box 605 can be understood with reference to the discussion of box 505a, the operation of box 610 with reference to the discussion of box 510a, the operation of box 615 with reference to the discussion of box 515a, the operation of box 625 with reference to the discussion of box 525a, the operation of box 630 with reference to the discussion of box 530a, the operation of box 635 with reference to the discussion of box 535a, the operation of box 640 with reference to the discussion of box 540a, the operation of box 645 with reference to the discussion of box 545a, the operation of box 650 with reference to the discussion of box 550a, the operation of box 655 with reference to the discussion of box 555a, the operation of box 660 with reference to the discussion of box 560a, and the operation of box 665 with reference to the discussion of box 565a. This article will focus on the operations of boxes 620a, 620b, 620c, and 620d, as special cases of the operations of box 520a.

[0174] box 605 and frame 610 - Initialization of a quantum computer Method 600 begins at box 605, where the classical computational system creates a complete Hamiltonian for the chemical system to be analyzed, and continues to box 610, so that the classical computational system generates an initial state for the chemical system and a parameterized quantum circuit representing the chemical system based on the initial state.

[0175] box 615 - Generate experimental wave function At box 615, the quantum computing system generates a parameterized wave function of the characteristic solution of the approximate complete Hamiltonian by executing parameterized quantum circuitry.

[0176] box 620a - Pauli sampling At box 620a, the quantum computing system performs Pauli sampling (for Pauli measurement) on the parameterized quantum circuit with respect to the complete Hamiltonian.

[0177] box 620b - Calculate the expected value At box 620b, the classical computing system calculates the expectation of the parameterized quantum circuit relative to the complete Hamiltonian.

[0178] box 620c - Update the parameters of the quantum circuit At box 620c, the quantum computing system updates a set of parameters of the parameterized quantum circuit based on the expected value.

[0179] box 620d - Determine if the third cycle is complete. At box 620d, the classical computing system determines whether a third stopping criterion has been met. In some such implementations, the third stopping criterion is met in response to at least one of the following: a predefined number of iterations in a third multi-iteration run has been performed; a predefined time has been reached for running the third multi-iteration run; a predefined time has been reached using a quantum computing system; the change in the expected value from a given iteration in the third multi-iteration run to subsequent iterations in the third multi-iteration run is within a threshold of the quantum noise change value of the quantum computing system; the first derivative of the expected value from a given iteration in the third multi-iteration run falls below a termination threshold; the change in the expected value from multiple previous iterations in the third multi-iteration run to subsequent iterations in the third multi-iteration run is within a termination threshold; and a predefined number of samples have been obtained across the third multi-iteration run. When a predefined number of samples is used as the third stopping criterion, each sample includes the following operations: obtaining a wavefunction by executing a quantum circuit via the quantum computing system; obtaining a measurement result by performing a measurement on the wavefunction via the quantum computing system; and outputting the measurement result.

[0180] In the box 620d The third stopping standard was not met. If the third stopping criterion has not yet been met, method 600 returns to box 620a so that the quantum computing system can continue to perform Pauli sampling or Pauli measurement on the now updated (according to box 620c) experimental wavefunction.

[0181] In the box 620d The third stopping criterion has been met. When the third stopping criterion is met, the quantum computing system terminates the third loop, and method 600 proceeds to block 625 with the output of the last iteration of the third multiple iterations for further processing according to blocks 625-665.

[0182] According to the diagram 7 Methods for determining ground state value and ground state energy Figure 7 This is a flowchart of an example method 700 for improving the computational efficiency of a hybrid quantum-classical computing system without using Pauli sampling or Pauli measurement to determine the ground state value and ground state energy of a Hamiltonian, according to an embodiment of this disclosure. The Hamiltonian can be used to represent an element or compound.

[0183] As a method 500a Special cases of methods 700 about Figure 7 The method of discussion 700 can be understood as being about Figure 5A The discussion focuses on a special case of method 500a, in which Pauli sampling or Pauli measurement relative to the parameterized quantum circuit is not performed. Accordingly, the operation of box 705 can be understood with reference to the discussion of box 505a, the operation of box 710 with reference to the discussion of box 510a, the operation of box 715 with reference to the discussion of box 515a, the operation of box 725 with reference to the discussion of box 525a, the operation of box 730 with reference to the discussion of box 530a, the operation of box 735 with reference to the discussion of box 535a, the operation of box 740 with reference to the discussion of box 540a, the operation of box 745 with reference to the discussion of box 545a, the operation of box 750 with reference to the discussion of box 550a, the operation of box 755 with reference to the discussion of box 555a, the operation of box 760 with reference to the discussion of box 560a, and the operation of box 765 with reference to the discussion of box 565a. This article will focus on the operations of box 720, as a special case of the operations of box 520a.

[0184] box 705 and frame 710 - Initialization of a quantum computer Method 700 begins at box 705, where the classical computational system creates a complete Hamiltonian for the chemical system to be analyzed, and continues to box 710, so that the classical computational system generates an initial state for the chemical system and a parameterized quantum circuit representing the chemical system based on the initial state.

[0185] box 720 Operation - Output parameterized wavefunction without Pauli sampling At box 720, the quantum computing system runs a third and multiple iterations via a hybrid quantum-classical computing system using a variable quantum feature solver (VQE) until a third stopping criterion is met. Each iteration of this third and multiple iterations includes generating a parameterized wavefunction of the ground state of an approximately complete Hamiltonian by executing parameterized quantum circuitry on the quantum computing system without performing Pauli sampling, and outputting the parameterized wavefunction.

[0186] Technical improvements provided by non-Pauli processes Unlike the third loop, which relies on a third stopping criterion to perform the analysis, the operation described in Method 700 relies on a second stopping criterion (evaluated according to box 725) and the sufficiency of the number of selected basis states (according to boxes 730-735) to identify when the quantum computer has produced a sufficiently accurate complete Hamiltonian representation to be presented in its subspace Hamiltonian. The Pauli-free method described in Method 700 allows hybrid quantum-classical computing systems to omit the computationally expensive operation of performing Pauli measurements on the quantum computing system, thus saving computational resources.

[0187] method 700 Continue operation Similar to combining about Figure 5A The discussion provided in Method 500a continues with Method 700 in boxes 725-765 until output is provided to the operator.

[0188] According to the diagram 8 Methods for determining characteristic solutions through operations Figure 8 This is a flowchart of an example method 800 for determining a characteristic solution of a Hamiltonian according to an embodiment of this disclosure, the Hamiltonian being used to represent a chemical system. Method 800 can be considered as relating to... Figure 5A The discussion focuses on special cases or variations of method 500a, where instead of using a single K value in each iteration (which may vary between iterations of various loops), multiple K values ​​of different sizes, denoted as k1, k2, ... k, are used in parallel. m , where m is the number of times the size of K is specified by the user.

[0189] method 800 Initialization Method 800 begins at box 805, where a complete Hamiltonian representing the chemical system is created. Method 800 then proceeds to one or more iterations of boxes 810-860 to solve for a subspace Hamiltonian of the chemical system using both quantum and classical computing resources. This subspace Hamiltonian represents the complete Hamiltonian in a classically tractable problem space. Because classical computing resources are more readily available than quantum computing resources, various techniques can be used to attempt handover multiple times in different iterations to make the computation tractable for classical computers, while still providing an improvement in overall system efficiency compared to determining the ground-state value and ground-state energy of the complete Hamiltonian entirely via quantum computing.

[0190] box 805 - Create a complete Hamiltonian At box 805, a complete Hamiltonian representing the chemical system is created using a classical computational system. The complete Hamiltonian has a value with 2... n The Hilbert space of 2^n basis states represents all possible states that the quantum system can exist in. This contrasts with the subspace Hamiltonian (discussed in detail in box 540a), which includes 2^n basis states chosen to best represent the properties of the chemical system. n A subset of basis vector states. The Hamiltonian is used to evaluate the energy of each state in Hilbert space, while the subspace Hamiltonian provides a computationally simpler platform to evaluate the energy state, which can be evaluated by classical computing systems. In contrast, the full Hamiltonian may be too complex for classical computing systems to process within a reasonable time frame, thus requiring the use of quantum computing systems for evaluation.

[0191] box 810 - Assign initial state At box 810, the classical computing system assigns an initial state to the complete Hamiltonian for the quantum computing system to begin iterative analysis. The initial state represents the entire set of 2... nThe initial allocation of amplitudes of the basis states is the starting point for computation. In various embodiments, classical computing systems can use various state preparation protocols to assign initial states to the complete Hamiltonian. These initial states can include at least one of the following: states prepared via the Hartree-Fock protocol, zero states, computational basis states in Hilbert space, states prepared via ab-initialization initial state preparation protocols (e.g., density functional theory (DFT), configuration interactions (CI), coupled clusters (CC), Møller-Plesset perturbation theory (MPn), etc.), states prepared via tensor network initial state preparation protocols (e.g., optimizing a tensor network and mapping the resulting states to quantum circuits to obtain quantum states), sparse initial states from the final iterative characteristic solutions of a previous set of iterative analyses from a hybrid computing system (e.g., the final iteration of previous iterations for subsequent iterations), uniformly distributed states, and randomly distributed states. This disclosure envisions that different protocols might be used when subsequent iterations are performed on the same chemical system by a quantum computing system.

[0192] From the box 810 To the box 810 The first loop iteration Different computations are performed using classical and quantum computing systems, and an evaluation iteration (and adjustments made thereto) based on the initial state is executed on a hybrid quantum-classical computing system. This evaluation is performed by running the first multiple iterations (e.g., as a loop) on the hybrid quantum-classical computing system until a first stopping criterion is met (as discussed with respect to box 850), and may include various sub-loops for different analyses.

[0193] box 815 - Generate experimental wave function At box 815, the quantum computing system generates the experimental wave function (||) describing the quantum state of an isolated quantum system based on known quantum mechanical practices. Ψ The Hamiltonian is analyzed through multiple iterations starting from the initial state to find an approximation of the characteristic solution of the Hamiltonian.

[0194] box 820 - Sampling test wave function At box 820, the quantum computing system samples the test wavefunction generated according to box 815 in the computational basis to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 In various implementations, sampling the test wavefunction updates the values ​​of various basis states based on the initial basis states (according to box 810) or early iterations of sampling the test wavefunction.

[0195] box 825 - Second cycle At box 825, the classical computing system determines whether a second stopping criterion has been met. In various implementations, the second stopping criterion is met in response to at least one of the following: a predefined number of iterations of the second loop from box 820 to box 825 have been performed; and the batch results contain at least a predefined number of values. When the second stopping criterion has not been met, method 800 returns from box 825 to box 820 so that the quantum computer can continue to perform additional iterations of the second loop. When the second stopping criterion has been met, method 800 proceeds from box 825 to box 830, thereby exiting the second loop.

[0196] box 830 - Post-processing At box 830, the classical computational system performs post-processing on the basis vector states to target the values ​​from k1 to k. m Each choice k obtains a set of criteria that satisfy the given conditions. S Each basis state is post-processed to evaluate each basis state according to a criterion to determine whether it should be used to construct a subspace Hamiltonian from the full Hamiltonian. In various implementations, the criterion is satisfied in response to at least one of the following: the second plurality of values ​​contains a predefined number of values ​​from the first plurality of values; the number of values ​​contained in the second plurality of values ​​is within a predefined threshold of the predefined number of values ​​from the first plurality of values; or the second plurality of values ​​satisfies one or more selection protocols, such as regarding... Figures 3A to 3J As described.

[0197] In the box 830 Multiple selection protocols are applied in China k 1-m Each of the m values ​​is different, therefore each of these m choices includes a different number of basis states. In various implementations, the selection protocol for choosing the basis states is different for k. i Each of them (e.g., for k) 1-m The i The members (k) can be the same or different, and various supplementary (or removal) protocols can be applied to make k 1-m Each k in i Select the corresponding number of basis states.

[0198] box 835 - Subspace Hamiltonian Construction At box 835, the classical computing system calculates for each selection k in group S according to a second plurality of values ​​selected from box 830, including criteria. i A subspace Hamiltonian is constructed to represent a chemical system. In various implementations, the subspace Hamiltonian representation includes at least one of the following: k i ×k i Matrices, Pauli sums, and black-box representations with element-wise access protocols. For the selected k... i The subspace Hamiltonian of each basis vector state can be expressed as: H' ki .

[0199] box 840 - Calculation of minimum eigenvalue and eigenvector At box 840, the hybrid quantum-classical computing system computes the subspace Hamiltonian. H' ki The characteristic solutions of the Hamiltonians in each subspace of the system. In various implementations, the classical computing system uses a characteristic solver to compute the characteristic solutions, which can be at least one of the following: a quantum computing characteristic solver that performs the computation on a quantum computing system, a classical computing characteristic solver that performs the computation on a classical computing system, or a hybrid computing characteristic solver that performs the computation on a hybrid computing system.

[0200] box 845 - Ground state value calculation At box 845, the classical computational system uses the eigenvalues ​​of the subspace Hamiltonian to compute approximate ground-state values ​​representing the ground state of the complete Hamiltonian, and approximate ground-state energy values ​​representing the ground-state energy of the complete Hamiltonian. The final eigenvalues ​​of the complete Hamiltonian are computed as the subspace Hamiltonian. H' ki The weighted sum of the eigenvalues ​​of the Hamiltonians in each subspace, where the weights are specified by the operator before method 800 begins, and the final eigenvalues ​​λ. F It can be represented as λ F = Σ i (wi*λ ki ), where wi is the subspace Hamiltonian H' ki eigenvalues ​​λ ki The weight.

[0201] box 850 - The first iteration terminates. At box 850, the classical computing system determines whether a first stopping criterion has been met, thus completing the computation loop. In various implementations, the first stopping criterion is met in response to at least one of the following: a predefined number of iterations in a first plurality of iterations has been performed, the first plurality of iterations have been run for a predefined time, the quantum computing system has been used for a predefined time, the change in the ground state energy value from a given iteration in the first plurality of iterations to subsequent iterations in the first plurality of iterations is within a threshold of the quantum noise change value of the quantum computing system, the first derivative of the ground state energy value from a given iteration in the first plurality of iterations falls below a termination threshold, and the change in the ground state energy value from multiple previous iterations in the first plurality of iterations to subsequent iterations in the first plurality of iterations is within a termination threshold.

[0202] box 850 Decision based on the first stopping criterion If the first stopping criterion has not yet been met, method 800 proceeds from box 850 to box 860 to determine how to continue based on various decision criteria. If the first stopping criterion has been met, method 800 proceeds from box 850 to box 855.

[0203] box 855 - Output At box 855, in response to the first stopping criterion being met and the classical computer stopping the first multiple iterations, the classical computing system outputs the characteristic solution of the final iteration of the first multiple iterations. In various implementations, once the characteristic solution is known, the operator can use it via the classical computing system to simulate a state relative to the target chemical system that can be represented as 2 n ×2 n The matrix corresponds to the complete Hamiltonian of the chemical system, and the most stable configuration and properties of the system are determined. Characteristic solutions of the chemical system provide information about its reactivity, stability, and potential interactions with other chemical systems. By evaluating the reactivity, stability, and interactions of the chemical system, the operator can use it to synthesize other chemical systems or study various reactions.

[0204] box 860 - Iterative determination At box 860, in response to the first stopping criterion not being met, the classical computing system determines which decision criteria have been met to determine how to continue. In various implementations, when the first stopping criterion has not been met, the operator is prompted to meet which of the three decision criteria to restart the computation fully or partially. In various implementations, method 800 can iterate multiple times from box 810 through box 860 until the first stopping criterion is met or the operator terminates method 800. In various implementations, each time method 800 iterates through box 860, at least one parameter in the computation is changed from the previous iteration to the next. For example, this could be done for each... Ki-m Choosing different thresholds allows for the selection of different procedures for determining the criteria, and different calculation methods can be used (e.g., method 500a, method 500c, etc.). In various implementations, the classical calculation system automatically determines which decision criteria to meet based on which loop is more likely to be repeated than others; while in other implementations, the classical calculation system prompts the operator (or has been pre-instructed by the operator) which decision criteria should be met.

[0205] In the box 860 The first decision criterion is met. When the first decision criterion has been met, method 800 returns from box 860 to box 810 to generate a new (e.g., second) initial state, and then re-runs the first multiple iterations on the hybrid quantum-classical computing system until the first stopping criterion is met in a subsequent (e.g., second) iteration.

[0206] In the box 860 The second decision criterion is met. When the second decision criterion is met, method 800 returns from box 860 to box 820, back to the node sampled from the experimental wavefunction, and then re-runs the first multiple iterations on the quantum computing system until the second stopping criterion is met in the subsequent iteration (e.g., the second time), and continues through boxes 820-860 until the first stopping criterion is met in the subsequent iteration (e.g., the second time).

[0207] In the box 860 The third decision-making standard is met. When the third decision criterion is met, method 800 returns from box 860 to box 830, reselecting values ​​for constructing the subspace Hamiltonian via a classical computing system. In various embodiments, the classical computing system uses the previously generated first plurality of values ​​and selects new second plurality of values, each of which satisfies the inclusion criterion. In various embodiments, the classical computing system may use different inclusion criteria or different processing protocols to select the new second plurality of values. Method 800 can then continue through boxes 830-850 until the first stopping criterion is subsequently met (e.g., a second time).

[0208] According to the diagram 9 Methods for determining characteristic solutions through operations Figure 9 This is a flowchart of an example method 900 for determining a characteristic solution of a Hamiltonian according to an embodiment of this disclosure, the Hamiltonian being used to represent a chemical system. Method 900 can be considered as relating to... Figure 5A The discussion focuses on a special case or variant of method 500a, in which the core space is no longer defined using a single hypothetical space in each iteration (which may vary between iterations of various loops), but rather the core space is expanded through various methods, including merging the current core space with the previous core space.

[0209] method 900 Initialization Method 900 begins at box 905, where a classical computational system is used to create a complete Hamiltonian representing the chemical system. The complete Hamiltonian has a value with 2... n The Hilbert space of _n basis states, representing all possible states the quantum system can exist in. This contrasts with the core space (discussed in detail in box 925), the subspace of the Hamiltonian, which includes 2_n basis states chosen to best represent the properties of the chemical system. n A subset of basis vector states. The Hamiltonian is used to evaluate the energy of each state in Hilbert space, while the subspace Hamiltonian provides a computationally simpler platform to evaluate the energy state, which can be evaluated by classical computing systems. In contrast, the full Hamiltonian may be too complex for classical computing systems to process within a reasonable time frame, thus requiring the use of quantum computing systems for evaluation.

[0210] box 910 - Assign initial state At box 910, the classical computing system assigns an initial state to the complete Hamiltonian for the quantum computing system to begin iterative analysis. The initial state represents the entire set of 2... n The initial allocation of amplitudes of the basis states is the starting point for computation. In various embodiments, classical computing systems can use various state preparation protocols to assign initial states to the complete Hamiltonian. These initial states can include at least one of the following: states prepared via the Hartree-Fock protocol, zero states, computational basis states in Hilbert space, states prepared via ab-initialization initial state preparation protocols (e.g., density functional theory (DFT), configuration interactions (CI), coupled clusters (CC), Møller-Plesset perturbation theory (MPn), etc.), states prepared via tensor network initial state preparation protocols (e.g., optimizing a tensor network and mapping the resulting states to quantum circuits to obtain quantum states), sparse initial states from the final iterative characteristic solutions of a previous set of iterative analyses from a hybrid computing system (e.g., the final iteration of previous iterations for subsequent iterations), uniformly distributed states, and randomly distributed states. This disclosure envisions that different protocols might be used when subsequent iterations are performed on the same chemical system by a quantum computing system.

[0211] box 915 - Generate experimental wave function At box 915, the quantum computing system generates the experimental wave function (||) describing the quantum state of an isolated quantum system based on known quantum mechanical practices. Ψ The Hamiltonian is analyzed through multiple iterations starting from the initial state to find an approximation of the characteristic solution of the Hamiltonian.

[0212] box 920 - Sampling test wave function At box 920, the quantum computing system samples the test wavefunction generated according to box 915 in the computational basis, in order to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 In various implementations, sampling the test wavefunction updates the values ​​of various basis states based on the initial basis states (according to box 910) or early iterations of sampling the test wavefunction.

[0213] box 925 - Finding the core space At box 925, the classical computational system determines the basis states from the computational basis samples (e.g., the proposed space) to form the initial core space for analysis. In various implementations, as described herein, different numbers of target basis states can be used for selection (K). K One or more selection protocols are used to determine the core space.

[0214] box 930 - Extend / Compress core space At box 930, the classical computational system may optionally expand, compress, or expand and compress the initial core space found in box 925. In various embodiments, the expansion space used to expand the initial core space may include a prior space, a core space identified from earlier iterations of method 900, or a deduced / generated set of basis states, such as basis states with α / β exchanges or basis states in Hamming space. These basis states forming the expansion space (typically or preferably) are located outside the proposed space from which the core space is derived and are selected as those basis states that, if included in subsequent computations, may significantly affect the accuracy of the characteristic solution. Furthermore, the basis states in the expansion space may undergo various selection protocols (alone, together with other expansion spaces, or together with the core space) to reduce the number of basis states subsequently used to construct the subspace Hamiltonian (e.g., according to box 935) and ensure that the selected basis states for analysis are (from the available basis states) the most significant basis states for computing accurate characteristic solutions.

[0215] Benefits of expanding and compressing core space Because the basis states in the extended space are part of the complete space of the chemical system, including additional basis states does not degrade the computational results for the characteristic solution of the chemical system; however, excessively including basis states with no or minimal impact may consume more computational resources than strictly required to reach the characteristic solution. Therefore, while extending the core space to include more basis states beyond those available in the currently proposed space can improve the overall results, classical computational systems can apply various selection protocols (as discussed in this paper) to compress the number of basis states in the initial core space, the extended space, or the extended core space to ensure that the number of basis states selected to be included in the subspace Hamiltonian does not exceed [a certain threshold]. K Furthermore, by using classical computational methods to identify these extended basis states, operators can reduce the amount of quantum computational resources required to generate larger or multiple hypothetical spaces (e.g., union hypothetical spaces) in multiple iterations.

[0216] box 935 - Subspace Hamiltonian Construction At box 935, the classical computational system constructs a subspace Hamiltonian from the basis vector states selected according to boxes 925 and 930 to represent the chemical system. In various embodiments, the subspace Hamiltonian representation includes at least one of the following: K × K Matrices, Pauli sums, and black-box representations with element-wise access protocols.

[0217] box 940 - Calculation of minimum eigenvalue and eigenvector At box 940, the hybrid quantum-classical computing system computes the characteristic solution of the subspace Hamiltonian. In various implementations, the classical computing system uses a characteristic solver to compute the characteristic solution, which can be at least one of the following: a quantum computing characteristic solver that performs the computation on a quantum computing system, a classical computing characteristic solver that performs the computation on a classical computing system, or a hybrid computing characteristic solver that performs the computation on a hybrid computing system.

[0218] box 945 - Ground state value calculation At box 945, the classical computational system uses the characteristic solutions of the subspace Hamiltonian to compute the approximate ground state value representing the ground state of the complete Hamiltonian, and the approximate ground state energy value representing the ground state energy of the complete Hamiltonian.

[0219] box 950 - The first iteration terminates. At box 950, the classical computing system determines whether a first stopping criterion has been met, thus completing the computation loop. In various implementations, the first stopping criterion is met in response to at least one of the following: a predefined number of iterations in a first plurality of iterations has been performed, the first plurality of iterations have been run for a predefined time, the quantum computing system has been used for a predefined time, the change in the ground state energy value from a given iteration in the first plurality of iterations to subsequent iterations in the first plurality of iterations is within a threshold of the quantum noise change value of the quantum computing system, the first derivative of the ground state energy value from a given iteration in the first plurality of iterations falls below a termination threshold, and the change in the ground state energy value from multiple previous iterations in the first plurality of iterations to subsequent iterations in the first plurality of iterations is within a termination threshold.

[0220] box 950 Decision based on the first stopping criterion If the first stopping criterion has not yet been met, method 900 proceeds from box 950 to box 960 to determine how to continue based on various decision criteria. If the first stopping criterion has been met, method 900 proceeds from box 950 to box 955.

[0221] box 955 - Output At box 955, in response to the first stopping criterion being met and the classical computer ceasing the first multiple iterations, the classical computing system outputs the characteristic solution of the final iteration of the first multiple iterations. In various implementations, once the characteristic solution is known, the operator can use it via the classical computing system to simulate a state relative to the target chemical system that can be represented as 2 n ×2 n The matrix corresponds to the complete Hamiltonian of the chemical system, and the most stable configuration and properties of the system are determined. Characteristic solutions of the chemical system provide information about its reactivity, stability, and potential interactions with other chemical systems. By evaluating the reactivity, stability, and interactions of the chemical system, the operator can use it to synthesize other chemical systems or study various reactions.

[0222] box 960 - Iterative determination At box 960, in response to the first stopping criterion not being met, the classical computing system determines which decision criteria have been met to determine how to continue. In various implementations, when the first stopping criterion has not been met, the operator is prompted to meet which of the four decision criteria to restart the computation fully or partially. In various implementations, method 900 can iterate multiple times from box 910 through box 960 until the first stopping criterion is met or the operator terminates method 900. In various implementations, each time method 900 iterates through box 960, at least one parameter in the computation is changed from the previous iteration to the next. For example, it could be... KBy selecting different thresholds, different procedures can be chosen for determining the criteria, etc. In various implementations, the classical computing system automatically determines which decision criteria to meet based on which loop is more likely to be repeated than others; while in other implementations, the classical computing system prompts the operator (or has been instructed in advance by the operator) which decision criteria should be met.

[0223] In the box 960 The first decision criterion is met. When the first decision criterion has been met, method 900 returns from box 960 to box 910 to generate a new (e.g., second) initial state, and then re-runs the first multiple iterations on the hybrid quantum-classical computing system until the first stopping criterion is met in a subsequent (e.g., second) iteration.

[0224] In the box 960 The second decision criterion is met. When the second decision criterion is met, method 900 returns from box 960 to box 920, back to the node sampled from the test wavefunction, and then re-runs the first multiple iterations on the quantum computing system until the second stopping criterion is met in the subsequent iteration (e.g., the second time), and continues through boxes 925-950 until the first stopping criterion is met in the subsequent iteration (e.g., the second time).

[0225] In the box 960 The third decision-making standard is met. When the third decision criterion has been met, method 900 returns from block 960 to block 925 to search for a new set of basis states via a classical computing system to define the initial core space. In various embodiments, the classical computing system uses the previously generated core space as a priori space to focus on or avoid the selection of new basis states within that space. In various embodiments, the classical computing system may use different inclusion criteria or different selection protocols to select new basis states. Method 900 can then continue through blocks 930-950 until the first stopping criterion is met subsequently (e.g., a second time).

[0226] In the box 960 The fourth decision criterion is met. When the fourth decision criterion has been met, method 900 returns from box 960 to box 930 to search for a new set of basis states via a classical computing system to define the expanded core space. In various implementations, the classical computing system may use either the initial core space (from box 925) or the expanded core space (from previous iterations of box 930) as a starting point, adding new basis states or reducing the number of basis states (e.g., removing those basis states that do not improve the accuracy of subsequent calculations). Method 900 can then continue through boxes 935-950 until the first stopping criterion is met subsequently (e.g., a second time).

[0227] According to the diagram 10A Methods for determining characteristic solutions through operations Figure 10A This is a flowchart of an example method 1000a for determining a characteristic solution of a Hamiltonian according to an embodiment of this disclosure, the Hamiltonian being used to represent a chemical system. Method 1000a can be considered as relating to... Figure 5A The special cases or variations of method 500a discussed here focus on defining the initial state rather than adjusting it in each iteration. K The value, the use of different selection protocols, the use of different selection criteria, or a combination thereof (these may vary between iterations of various loops).

[0228] method 1000a Initialization Method 1000a begins at box 1005, where a classical computational system is used to create a complete Hamiltonian representing the chemical system. The complete Hamiltonian has a value with 2... n The Hilbert space of _n basis states, representing all possible states the quantum system can exist in. This contrasts with the subspace Hamiltonian (discussed in detail in box 1030), which includes 2_n basis states chosen to best represent the properties of the chemical system. n A subset of basis vector states. The Hamiltonian is used to evaluate the energy of each state in Hilbert space, while the subspace Hamiltonian provides a computationally simpler platform to evaluate the energy state, which can be evaluated by classical computing systems. In contrast, the full Hamiltonian may be too complex for classical computing systems to process within a reasonable time frame, thus requiring the use of quantum computing systems for evaluation.

[0229] box 1010 - Assign initial state At box 1010, the classical computing system assigns an initial state (|Φ〉) to the complete Hamiltonian for the quantum computing system to begin iterative analysis. The initial state represents the entire set of 2. nThe initial allocation of amplitudes of the basis states is the starting point for computation. In various embodiments, classical computing systems can use various state preparation protocols to assign initial states to the complete Hamiltonian. These initial states can include at least one of the following: states prepared via the Hartree-Fock protocol, zero states, computational basis states in Hilbert space, states prepared via ab-initialization initial state preparation protocols (e.g., density functional theory (DFT), configuration interactions (CI), coupled clusters (CC), Møller-Plesset perturbation theory (MPn), etc.), states prepared via tensor network initial state preparation protocols (e.g., optimizing a tensor network and mapping the resulting states to quantum circuits to obtain quantum states), sparse initial states from the final iterative characteristic solutions of a previous set of iterative analyses from a hybrid computing system (e.g., the final iteration of previous iterations for subsequent iterations), uniformly distributed states, and randomly distributed states. This disclosure envisions that different protocols might be used when subsequent iterations are performed on the same chemical system by a quantum computing system.

[0230] box 1015 - Generate experimental wave function At box 1015, the quantum computing system generates the experimental wave function (||) describing the quantum state of an isolated quantum system based on known quantum mechanical practices. Ψ The Hamiltonian is analyzed through multiple iterations starting from the initial state to find an approximation of the characteristic solution of the Hamiltonian. An experimental wavefunction is prepared based on the initial state |Φ>.

[0231] box 1020 - Sampling test wave function At box 1020, the quantum computing system samples the test wavefunction generated according to box 1015 in the computational basis to obtain a value from 2... n A set of basis states is obtained N basis states, where each basis state b j The probability of being sampled is | α j | 2 In various implementations, sampling the test wavefunction updates the values ​​of various basis states based on the initial basis states (according to box 1010) or earlier iterations of sampling the test wavefunction. Quantum computers can repeatedly perform projection measurements within a computational basis. N s Second-rate.

[0232] box 1025 - Post-processing At box 1025, the classical computing system performs post-processing on the basis states to identify which basis states are used when constructing the subspace Hamiltonian (according to box 1030). In various implementations, the classical computing system computes each integer (configuration).i In N s The frequency (f) of occurrence in the next emission result can be measured as f i = n i / N s , where n i yes i In the measurement results i1, ..., i Ns The number of times it appears. Based on frequency measurements, the classical computing system selects... K The most frequently occurring configurations k1, k2, k3, ..., k K Let ∈ {0, 1, ..., 2n - 1}, and define a subspace S = span{|k1〉, ..., |k K 〉}.

[0233] box 1030 - Subspace Hamiltonian Construction At box 1030, the classical computational system constructs a subspace Hamiltonian from the basis vector states selected in box 1025 for subspace S to represent the chemical system. In various embodiments, the subspace Hamiltonian representation includes at least one of the following: K×K Matrices, Pauli sums, and black-box representations with element-wise access protocols.

[0234] box 1035 - Calculation of characteristic solutions At box 1035, the hybrid quantum-classical computing system computes the characteristic solution of the Hamiltonian of the subspace. In various implementations, the classical computing system uses a characteristic solver to compute the characteristic solution, which can be at least one of the following: a quantum computing characteristic solver that performs the computation on a quantum computing system, a classical computing characteristic solver that performs the computation on a classical computing system, or a hybrid computing characteristic solver that performs the computation on a hybrid computing system. The classical computing system can perform a selected CI computation or diagonalize the effective Hamiltonian in the subspace S to give an approximate ground state and ground state energy of the Hamiltonian from which the value of the complete Hamiltonian can be derived.

[0235] box 1040 - Determine if the solution converges At box 1040, the classical computational system determines whether the characteristic solutions (calculated according to box 1035) of the subspace Hamiltonian and the complete Hamiltonian converge within a convergence threshold. When the characteristic solution is determined to be convergent, method 1000a proceeds to box 1055. Otherwise, when the characteristic solution is determined not to be convergent, method 1000a proceeds to box 1045.

[0236] box 1045 - New state preparation At box 1050, the classical computational system is based on θ calculated in the last executed iteration of box 1010 or box 1045. KThe value is evaluated via a quantum computing system to prepare a new state, as in... Figure 10B The discussion is as follows regarding method 1000b. Then, method 1000a can return to box 1020 to evaluate the new state and can continue until the feature solution converges (according to box 1040) or another stopping criterion is met.

[0237] box 1050 - Output At box 1050, in response to the classical computing system determining that the characteristic solution has converged, the classical computing system outputs the characteristic solution of the final iteration. In various implementations, once the characteristic solution is known, the operator can use it via the classical computing system to simulate a state relative to the target chemical system that can be represented as 2 n ×2 n The matrix corresponds to the complete Hamiltonian of the chemical system, and the most stable configuration and properties of the system are determined. Characteristic solutions of the chemical system provide information about its reactivity, stability, and potential interactions with other chemical systems. By evaluating the reactivity, stability, and interactions of the chemical system, the operator can use it to synthesize other chemical systems or study various reactions.

[0238] method 1000a frame 1010 , 1045 Detailed operation of input state selection Method 1000a relies on the selection of input states containing important configurations (e.g., according to box 1010 or box 1045), thus with extremely large weights |αi| 2 Describes the precise ground state. Therefore, Figure 10B The flowchart illustrating method 1000b provides detailed operations for blocks 1010 or 1045 of method 1000a when used to adaptively construct the input state. In method 1000b, the classical computing system evaluates state e. iθPj |c v The gradient h of the expected energy value j Gradient h j The value can be obtained via classical computation systems by using the Hermitian operator i[H, P] j Projected onto subspace S v And calculate the classical vector c v The expected value is used for calculation.

[0239] method 1000b The target of input state selection Ideally, the exact ground state |ψ| of the Hamiltonian H itself GS > is a candidate for this type of input state because |ψ GS Weights of important configurations in > |αi| 2 Typically, the weight is greater than that of less important configurations. One can measure this by considering |ψ GSProjection measurements are performed to select important configurations. Therefore, quantum states with lower energy expectations <Φ|H|Φ> can be used as input states, and they are expected to resemble exact ground states.

[0240] box 1055 - Define operator pool At box 1055, the system defines the operator pool ℙ = {P1, ..., P}. T} and the initial state |ψ0〉, and the iteration value v Initially set to zero (e.g., v = 0). In various implementations, the operator pool comprises a single Pauli operator.

[0241] box 1060 - Selecting Operators At box 1060, select the pool with the largest gradient value |h j The operator | is denoted as P tv If | h tv | If the value is less than the stopping threshold, method 1000b can stop and continue according to the relevant information. Figure 10A The method discussed is described in box 1015 or box 1020 of method 1000a. In method 1000a The current θ selected from the last iteration of box 1065 in method 100b is used. v The value as θ K .

[0242] box 1065 - Parameters for determining the new input state At box 1065, the classical computer determines the parameters of the new input state. In various implementations, the parameters are chosen to minimize or reduce relative to θ. v Energy expectation f v (θ v ) =〈c v | e -iθvPtv H e iθvPtv |c v This determination can be performed by a classical computing system because... P tv It is to satisfy P 2 tv = Pauli operator for I. v (θ v The value of ) for any θ v The values ​​can all be evaluated by H, P tv H P tvand i[H, P tv Projected onto subspace S v And calculate them for state |c v The expected value of f > can be efficiently executed by a classical computer. v (θ v f(x) is a simple trigonometric function, so a classical computer can calculate f(x) from these expected values. v (θ v The exact minimum value of ).

[0243] box 1070 - Define the state and iterate the process At box 1070, the state is defined as |Φ v +l〉eiθk*Ptk |Φ0〉, and v The value is iterated (e.g., v = v +1), and method 1000b uses v The new value is returned to box 1060.

[0244] picture 11A To the diagram 11F Example energy calculations in Figures 11A to 11F Examples illustrate different choices when constructing the subspace Hamiltonian according to embodiments of this disclosure. K The energy of a chemical system. In Figures 11A to 11F In each of these, the energy value of the chemical system calculated by VQE is shown as a data point represented by a square. The energy value of the chemical system, representing the ideal value of the energy level calculated by full configuration interaction (FCI), is shown as a data point represented by "x". The energy values ​​calculated for various transfer settings are shown as circles.

[0245] With noise model Li 2 S energy Figures 11A to 11C An example is given of the analysis of the chemical system of lithium sulfide (Li2S), whose quantum state was developed using a quantum computer simulator with 10 qubits, six electrons, an STO-3G basis set, and a noise model.

[0246] No noise model Li 2 S energy L without a noise model i2 S energy Figures 11D to 11F Examples are given for lithium sulfide (L i2 The analysis of the chemical system of S) whose quantum state was developed using a quantum computer simulator with 10 qubits, six electrons, STO-3G basis set and without applying a noise model.

[0247] picture 11A To the diagram 11F Description of the charts in Regardless of whether a noise model was applied, and regardless of... K Regardless of the chosen value, the values ​​calculated solely by the quantum computing simulator system showed lower accuracy compared to those calculated via transfer. For example... Figures 11A to 11F As shown, when applying the noise model, the calculated energy values ​​from the conventional VQE range from approximately -406.25 Ha to -406.75 Ha, while without the noise model, they range from -406.25 Ha to -406.6 Ha. However, when applying the noise model, the calculated energy values ​​from the transfer range from approximately -407.8 Ha to -408.0 Ha, while without the noise model, they range from -407.75 Ha to -408.0 Ha. Compared to the differences shown by the transfer calculations, the VQE calculations consistently show a larger difference from the ideal FCI value (and are therefore less accurate).

[0248] Accuracy improvement This improvement in accuracy is at least partly attributable to the fact that the classical computational system taking over the computation is unaffected by quantum noise. This is because the subspace Hamiltonian is sampled from states prepared by the VQE. K Constructed from individual basis vector states, classical computational systems can utilize the initial accuracy (and speed) of VQE as input, and then perform computations in an environment unaffected by quantum noise to achieve more accurate characteristic solutions, the accuracy of which might otherwise be limited by the amount of noise. Therefore, even... K At lower values, hybrid computing systems using handover also outperform traditional VQE systems. However, with... K As the value increases, the output of the hybrid system using the transfer approaches the accuracy provided by the FCI system.

[0249] handover procedures Beneficially, because the subspace Hamiltonian is used K Each chosen basis state is constructed from the matrix elements of the complete Hamiltonian, so the minimum energy of the subspace cannot be lower than the actual minimum energy. Therefore, even at higher... K Values ​​typically produce more accurate characteristic solutions (because as...) K With increasing scalar density, classical methods would approximate an exact solution, but the transfer method allows for solutions even at lower scalar density. K Even with values ​​below a certain threshold, the "upper limit" eigenvalue can be reliably generated (where the actual eigenvalue is known to be equal to or less than the generated eigenvalue), which still provides useful data. Although increasing the value in general... K It improves the accuracy of the characteristic solutions, but does not increase the number of basis states that are important to the characteristic solutions; on the contrary, it increases... KIt simply increases the likelihood that these states will be included in the selection. Therefore, by combining this with a very efficient selection protocol, the operator can construct selections with much smaller... K The subspace Hamiltonian also provides highly accurate characteristic solutions, as shown experimentally (see, for example, [reference needed]). Figures 11A to 11F Even these "upper limit" characteristic solutions are usually more accurate than those produced by other computationally more expensive operations, such as quantum VQE.

[0250] When to carry out the handover procedure The transfer procedure is executed after VQE, but the exact timing of the transfer is not well-defined. The transfer procedure requires the ability to sample basis states (bit strings) from the quantum state prepared by VQE (or any other quantum computing method). The more relevant basis states sampled, the better the transfer procedure performs. This exact point in time is usually unclear; however, there is no limit to the number of transfer attempts, so one can perform the first step of the transfer (i.e., the sampling of basis states) and determine if enough basis states have been sampled to successfully execute the remaining transfer steps. The first step of the transfer is the least resource-intensive. One can also execute the entire transfer procedure, and if the resulting solution is not good enough, one can resume quantum state optimization from where it left off and apply the transfer again later.

[0251] Benefits of the transfer By performing the handover operations described in this disclosure, operators are given greater flexibility in inspecting and modifying chemical systems. Operators can select several points in the process for iteration, allowing them to selectively conserve computational resources to produce more accurate characteristic solutions than conventional computational procedures.

[0252] Classical computing devices Figure 12 A classical computing device 1200 according to an embodiment of the present disclosure is illustrated. The classical computing device 1200 may include a processor 1210, a memory 1220, and a communication interface 1230.

[0253] processor Processor 1210 can be any processing unit capable of performing the operations and programs described in this disclosure according to instructions or directives. In various embodiments, processor 1210 can represent a single processor, multiple processors, a processor with multiple cores, and combinations thereof.

[0254] memory Memory 1220 is a device that may be volatile or non-volatile memory and may include RAM, flash memory, cache, disk drives, and other computer-readable storage devices. Although shown as a single entity, memory 1220 may be divided into different memory storage elements, such as RAM and one or more hard disk drives. As used herein, memory 1220 is an example of a device that includes computer-readable storage media and should not be construed as a transmission medium or the signal itself.

[0255] The memory contains executable instructions. As shown in the figure, memory 1220 includes various instructions executable by processor 1210 to provide operating system 1222 with various functions to manage classical computing device 1200, and to provide one or more programs 1224 with various functions to users of classical computing device 1200, including one or more of the features and functionalities described in this disclosure.

[0256] Programming selection does not require excessive experimentation. Those skilled in the art will recognize that different approaches can be taken in selecting or designing the program 1224 for performing the operations described herein, including the choice of programming language, the operating system 1222 used by the classical computing device 1200, and the architecture of the processor 1210 and memory 1220. Therefore, those skilled in the art will be able to select or design a suitable program 1224 based on the details provided in this disclosure.

[0257] Peripheral equipment Communication interface 1230 facilitates communication between the classical computing device 1200 and other devices, such as those mentioned above. Figure 12 The computing device described. In various embodiments, the communication interface 1230 includes an antenna for wireless communication and various wired communication ports. The classic computing device 1200 may also include or communicate via the communication interface 1230 with one or more input devices (e.g., a keyboard, mouse, stylus, touch input device, etc.) and one or more output devices (e.g., a display, speaker, printer, etc.).

[0258] Connect computing devices together Despite Figure 12 Not explicitly shown, but it should be recognized that the classic computing device 1200 can be connected to one or more public and / or private networks via a suitable network connection through the communication interface 1230. It will also be recognized that software instructions can also be loaded from a suitable storage medium or via wired or wireless means into a non-transitory computer-readable medium, such as memory 1220.

[0259] A restatement of classical computing devices Therefore, the classical computing device 1200 is an example of a system including a processor 1210 and a memory 1220, the memory including instructions (when executed by the processor 1210) for executing various embodiments of the present disclosure. Similarly, the memory 1220 is a means including instructions that, when executed by the processor 1210, execute various embodiments of the present disclosure.

[0260] quantum computing devices Figure 13 A quantum computing device 1300 according to an embodiment of the present disclosure is illustrated. The quantum computing device 1300 may include at least a communication interface 1310, a quantum programming interface 1320, a quantum state preparation circuit 1330, a quantum computing circuit 1340, and a quantum control and measurement circuit 1350.

[0261] Communication interface Communication interface 1310 facilitates communication between quantum computer 1300 and other devices, which may include classical computers (e.g., such as those related to quantum computers). Figure 12 The classical computing device 1200 under discussion (when it acts as a client to access the quantum computing device 1300 for quantum applications). In various implementations, the communication interface 1310 includes an antenna for wireless communication and various wired communication ports. The classical computer preprocesses input data (e.g., parameters, configurations, etc.) and transmits it to the quantum computing device 1300, retrieves and post-processes output data from the quantum computing device 1300, and finally interprets and presents the results to the end user.

[0262] quantum computer Quantum computers are designed to solve problems that classical computers cannot efficiently solve by utilizing the laws of quantum mechanics. Although still in their early stages, quantum computers show immense promise for a multitude of potential applications, including cryptography, artificial intelligence, drug discovery, materials science, scientific research and simulation, and countless others. Quantum computers and classical computers differ fundamentally in their basic principles, computational models, and capabilities. Classical computers process information in bits, each bit representing only one of two binary states at a time using deterministic logic; in contrast, quantum computers process information in multiple qubits, each of which can simultaneously represent a coherent superposition of two binary states. Furthermore, two or more qubits can become entangled, resulting in highly correlated quantum states that enable large-scale quantum parallelism, allowing a large number of operations to be performed concurrently. Quantum computers can also utilize quantum interference to amplify the probability of obtaining correct results and suppress the probability of obtaining incorrect results. Compared to classical computers, quantum computers have the potential to achieve exponentially faster computation speeds and solve extremely complex problems with high accuracy, exceeding the capabilities of today's most powerful (classical) supercomputers.

[0263] Quantum computer hardware The hardware used to provide qubits may vary in different implementations to match the application requirements of a given quantum computing device. In some implementations, qubits encode quantum states via photons (polarization), coherent states of light, electrons (spin), atomic nuclei of atoms or compounds, trapped ions, quantum dots, etc., many of which are still under development. Therefore, the illustrated quantum computing device 1300 is provided as a general representation of a quantum computing device 1300 that can implement qubits and influence state changes therein via various hardware technologies.

[0264] Context Despite Figure 13 While not explicitly shown, this disclosure envisions that the quantum computing device 1300 can be connected to one or more public and / or private networks via a suitable network connection through a communication interface 1310. This disclosure also envisions that software instructions can be loaded from a suitable storage medium or via wired or wireless means into a non-transitory computer-readable medium.

[0265] Quantum Programming Interface The quantum programming interface 1320 defines a common quantum programming interface for different clients through a set of application programming interfaces (APIs) created by quantum programming software. The quantum programming interface 1320 provides a representation of the underlying quantum hardware details, enabling clients to access the resources of the quantum computing device 1300 (e.g., client-agnostic quantum computing) independently of specific hardware implementations. Figure 13 As shown, quantum inputs are generated at the start of quantum processing using quantum programming interface 1320. These inputs represent data that is correctly formatted and suitable for further quantum computing. At the end of quantum processing, the quantum output data is converted into a client-readable format and transmitted to the client via communication interface 1310.

[0266] Quantum state preparation circuit The quantum state preparation circuit 1330 further encodes the quantum input to generate quantum states, which represent the initial states of the qubits, such as superposition, entanglement, probabilistic interpretation, and continuous evolution. Quantum states encode the information and properties of the quantum system, govern the behavior of physical components, and form the basis of quantum algorithms.

[0267] Quantum computing circuits The quantum computing circuit 1340 implements quantum algorithms and performs quantum computations using quantum state-driven quantum gates. The quantum computing circuit also includes a quantum memory 1342, a quantum processing unit 1344, and quantum error detection and correction 1346.

[0268] Quantum memory The quantum memory 1342 stores and preserves multiple quantum states in various superpositions for a period of time. The quantum processing unit 1344 is an indispensable component, operating based on quantum computer principles to perform quantum mechanics-based tasks. The quantum processing unit 1344 also stores computational states in the form of quantum mechanical states and communicates with various other units of the quantum computer 1300 using a quantum bus. The quantum error detection and correction unit 1346 locates and corrects errors that occur during quantum computing operations due to noise and decoherence.

[0269] Quantum control and measurement circuits like Figure 13 As shown, the quantum control and measurement circuit 1350 controls and monitors the operation of the quantum state preparation circuit 1330 and the quantum computing circuit 1340 to assist in the error detection and correction process. At the end of the quantum computing, the quantum control and measurement circuit 1350 measures the results so that the quantum output can be prepared to be converted into a client-readable format via the quantum programming interface 1320 and provided to the client via the communication interface 1310.

[0270] Overall operation Therefore, the quantum computer 1300 is a system example that includes a quantum programming interface 1320 and a set of quantum circuits (1330, 1340 and 1350) for performing complex quantum computing tasks received from a client via a communication interface 1310.

[0271] Description of quantum computing System energy Given the Hamiltonian H and the state |ψ〉, numerical values ​​can be calculated on a quantum computer. e The value of . When H gives a description of the system's energy, and |ψ> the system's state, the calculated value is . e This gives the energy of the system as defined by H in the state |ψ〉. This is called the expectation of |ψ〉 in H, and is mathematically written as Equation 1.

[0272] <ψ|H|ψ> [Formula 1] State and Theta (θ) Relationship On a quantum computer, a state |ψ(θ)〉 can be created, where the value of the state, or more precisely, the vector defining the state, depends on the values ​​of some parameters given by θ. The minimization of the expected value of |ψ(θ)〉 over H with respect to the parameter θ can be expressed by Equation 2.

[0273] min θ 〈ψ(θ)|H|ψ(θ)〉[Formula 2] formula 2 Explanation In Equation 2, the value of θ is chosen to minimize <ψ(θ)|H|ψ(θ)>, or more precisely, to minimize the energy of system H in state |ψ(θ)>. This is typically how the Variational Quantum Characteristic Solver (VQE) works; that is, updating the value of θ to minimize energy. e The final result is the minimum energy value. e (What VQE can achieve) and the value of θ (θ is a parameter vector). Once these parameters are obtained, the state |ψ(θ)> can be easily recreated for use in further algorithms.

[0274] State as a vector A quantum computer creates a state |ψ(θ)〉, which can be defined as a vector whose complexity (length) is exponentially related to the number of qubits. Quantum devices are very efficient at creating and storing these large vectors (e.g., relative to classical computers), which is why they are so powerful in performing these computations. Even for a moderate number of qubits, these vectors are beyond the storage or processing capabilities of classical computers. The vector is "stored" in a quantum superposition state, so it is impossible to view the vector in any other way than by sampling.

[0275] use 4 Example state of one qubit For example, when processing nWhen there are 4 qubits, the state |ψ(θ)〉 on the quantum computer can be represented as a vector, as shown in Equation 3.

[0276] |ψ(θ)〉= α1 |0000〉+ α2 |0001〉+ α3 |0010〉+ ... + α2 n |1111〉[Formula 3] Vectors |0000〉, |0001〉, ..., |1111〉 are called basis states, and any state (or vector) can be constructed as a linear combination of these states, and these basis states can be represented by so-called "one-hot vectors". For example, in Equation 4: [Formula 4] Minimum Energy Assumption The "one-hot vector" can be used to represent the state shown in Equation 5. The true minimum energy that can be given by > (where GS refers to the ground state or the true minimum energy state) is the state that is expected to be created, and can be represented in vector form as shown in Equation 6.

[0277] | 〉= ω1|0000〉+ ω2|0001〉+ ω3|0010〉+ ... + ω2 n |llll〉[Formula 5] [Formula 6] Quantum state sampling When sampling a quantum state (as shown in Equation 3), according to Born's rule, only a single basis state |b is returned. i For example, the returned state could be state |0010>, with a probability of |α. i | 2 For example, |α3| 2 From repeated sampling, the operator can see which basis states have the largest magnitude of α. i value.

[0278] Obstacles to state creation However, in practice, creating states like those shown in Equation 5 on a quantum computer presents many obstacles. These obstacles include the general difficulty of optimization (local minima) and the noise level on quantum devices. Assume the quantum computer converges to the state |ψ(θ*)〉, and the task of the quantum computer is to further refine this state towards the true minimum energy state |ψ(θ*)〉. >Optimization, most ω i The value is 0, or very close to 0, so that setting these values ​​to 0 has almost no effect on the energy value achieved by the state.

[0279] Assume the states are sparse. Therefore, by using the assumption that the state is sparse (or that the state can be well approximated by sparse states), the problem can be shifted to determining which ω i The value is significantly nonzero (or has other significant meaning), then the element is determined / selected. b i > Solve for the correct ω i value.

[0280] Sampling is used to identify states with significant non-zero amplitudes. Since quantum computers have created a well-approximate state |ψ(θ*)〉, classical computers can use this state to extract the necessary information to continue computation using classical computing resources by initially running algorithms such as VQE. Classical computers sample this state to obtain information about which basis states are most likely to have significantly non-zero ω. i Information about the values ​​(these ω) i The value can also be referred to as amplitude. Theoretically, an operator can sample any number of basis states, and generally, the larger the sample size, the more accurate the results, because a larger sample size increases the probability of capturing basis states with non-zero amplitudes. However, in practice, operators are usually limited by time and space (e.g., qubits) in the number of samples that can be collected, especially for larger problems.

[0281] Sampling generates a representative approximation of sparse states. Assume |ψ(θ*)〉 is | Since the approximation of the state |ψ(θ*)> is a good approximation, and the probability of sampling a particular basis state is given by the square of the magnitude of that state, then those basis states with the largest magnitudes will have the largest probability of being sampled during the sampling operation. Therefore, with a sufficient number of samples, it is reasonable to identify all basis states with sufficiently large magnitudes in the approximate state |ψ(θ*)> created by the quantum computer. As will be understood, these samples are qualitative rather than quantitative, because only obtaining the values ​​of the basis states is important; the statistics of the sampling and the specific methods used in the sampling are relatively minor, although the statistics may play a role in some adaptations of the algorithm, such as sorting or filtering basis states when needed.

[0282] The significance of the handover When |ψ(θ*)〉 is | When a good approximation of || is found, a quantum computer can delegate operations to a classical computer because the reduction in the number of basis states makes the computation possible for the classical computer, while the values ​​of the basis states are provided by the quantum computer. This delegation is effective because quantum computers excel at finding basis states with large amplitudes, but noise and quantum uncertainty in (current) quantum computers prevent them from reliably providing the values ​​for ||. The precise amplitude value of >. Therefore, after quantum computers have overcome the limitations of classical computers, classical post-processing comes into play at this point to overcome the limitations of quantum computers.

[0283] Reduce the full Hamiltonian to a subspace Hamiltonian. Once a set of basis states has been sampled, the Hamiltonian H is then reduced to a subspace Hamiltonian that includes only the matrix elements associated with these basis states, and the problem's dimensionality is simplified to a level sufficient for classical solution. The order of the basis states is not important when constructing the subspace Hamiltonian, as long as the order of the selected rows and columns remains consistent between the subspace Hamiltonian and the full Hamiltonian H.

[0284] Classical solutions are superior to quantum solutions. The classical solution obtained through the transfer output is indeed an approximation of the exact solution, but the energy given by this approximation is closer to the actual solution than the energy that could be obtained using the quantum device alone. In other words, the quantum device is used to help simplify the problem to only those basis states that are significant, and then, considering only these significant basis states, a very good approximate solution can be found using the classical method. It has also been shown that the minimum energy of this subspace cannot be lower than the actual minimum energy.

[0285] Classic Enhancement VQE Overview As will be understood, classical enhancement of VQE can be summarized as a method that enhances the performance of quantum states generated by VQE by incorporating additional classical states into the computation, and allows the target ground state to be given as a combination of quantum states and (optionally multiple) classical states. The handover procedure provided in this disclosure may also be classically enhanced, or may include basis state values ​​provided by other third parties, but as will be apparent upon a detailed reading of this disclosure, it differs from conventional classical enhancement methods.

[0286] Representation of the classical augmented solution For example, Equation 8 can represent the classical augmented solution, where | > is a classic enhanced solution, | > is a quantum state, and |C i > is a classical state used to enhance the solution.

[0287] | >≈ | >= σ0 | 〉+ σ1 |C1〉+ σ2 |C2〉+ ... + σ k |C k )[Formula 8] transfer VQE Overview In contrast, the transferable VQE can be summarized as follows: This method assumes that the ground state can be well approximated by sparse states (an assumption not made in classical enhanced VQE), and then uses the quantum states optimized by VQE to sample which states might have non-negligible contributions. The ground state problem is then simplified to a classically tractable problem on a classical computer using labels (basis states) of the non-negligible states, and the approximate ground state is solved using classical methods. The problem becomes classically tractable because it approximates the ground state using only those sampled states, thus reducing the problem size in terms of time and memory requirements.

[0288] Representation of transfer solutions on quantum computers The original VQE problem is creating states on quantum devices. As shown in Formula 5.

[0289] This state must be created on a quantum device because the problem is classically intractable. (VQE creation |) The quantum computer samples some basis states from this set, forming a group {|b1〉, |b2〉, ..., |bk〉}. An example of this group could be {|0011〉, |1010〉, |0101〉}, and then constructs the ground state as shown in Equation 9.

[0290] | 〉≈ |ψHandOver〉= σ1 |b1〉+ σ2 |b2〉+ ... + σ k |b k )[Formula 9] transfer VQE Compared to classic enhancement VQE Improvements provided Therefore, although the transfer may involve VQE, similar to classical augmentation solutions, the use of sparse states of sampled non-zero (or otherwise significant) basis vector states uniquely allows computations in hybrid computing systems to be transferred between quantum and classical computers. This improves the overall system performance, reduces the computational resources required by traditional computing systems, and increases the speed and accuracy of the performed computations, among other technical improvements and benefits.

[0291] transfer VQE Differences in state preparation on quantum devices Transfer VQE does not prepare a rough approximation of the target state on a quantum device, but rather prepares a state that provides a good distribution of important basis states from which to sample. To illustrate this point about the ground state, a satisfactory state for the transfer VQE method is... K The equal-amplitude superposition of several important basis states facilitates the sampling of these important basis states. Traditional analytical methods teach against using this state because it is considered not a good approximation of the ground state.

[0292] transfer VQE Differences offered in terms of control VQE transfer allows operators to apply selection heuristics to further narrow down candidate basis vector pools, rather than simply controlling the number of electrons and total spin—something not previously developed in the art. Furthermore, several control loops are defined during the VQE transfer process, allowing feedback to be incorporated into the analysis. Unlike being confined to building (e.g., gradually increasing in size) a quantum circuit to the state prepared for analysis or restarting the analysis from scratch, these control loops in the transfer VQE process allow the quantum circuit structure used to remain unchanged. Instead, the parameters of the quantum circuit can be modified with each iteration, as can the initial state on which the quantum circuit acts.

[0293] transfer VQE Improvements compared to other hybrid methods By performing the transfer VQE described in this disclosure, hybrid systems can adjust the computational load between quantum and classical computing systems through a rapid iterative process. Results obtained in earlier iterations can be reused in later iterations, further improving analysis speed and saving computational resources. Unlike other methods, in transfer VQE, when the initial selection includes the number of basis states in the subspace Hamiltonian (… K The value of is considered too computationally demanding for classical computing systems to solve, or when the operator chooses to use different selection protocols to determine which basis states in the complete Hamiltonian should constitute the subspace Hamiltonian. K When dealing with each basis vector state, no manual operator intervention is required to reset the hybrid system. Because these choices are made via classical computing using data already processed by the quantum computing system, iteration can be performed using classical methods, even though different characteristic solutions can be produced, thus further conserving quantum computing resources while checking or testing the solution results.

[0294] Additional understanding of the combination of implementation schemes In addition to the above-described embodiments, many other examples of specific combinations also fall within the scope of this disclosure, some of which are detailed below: Terms and Conditions 1 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K A set of basis states, wherein: the proposed space consists of a subset of basis states from the complete basis space of the chemical system, the basis states being sampled from a single parameterized quantum circuit on a quantum computer system, the complete basis space consisting of each basis state describing the chemical system, and the basis states being selected according to a selection protocol. K A number of basis vector states are used to define the core space of the chemical system, wherein the selection protocol uses, in addition to fixed vector states, to define the core space of the chemical system. K In addition to the maximum value selection protocol, at least one selection criterion identifies each basis state to be included in the core space from the proposed space, in the fixed... K In the maximum value selection protocol, each basis state is selected as the K highest probability basis state in the proposed space based on its probability of being among the top K highest probabilities. K One of the basis vector states; calculates the characteristic solution of the chemical system from the core space via a characteristic solver; and outputs the characteristic solution of the chemical system.

[0295] Terms and Conditions 2 : The method according to any one of clauses 1 to 22, wherein the selected K Each basis vector state also includes: selecting a set of initial base vectors from the proposed space. K selected Each basis vector state defines a first portion of the core space of the chemical system, and is fewer in number than a set defining the core space. K desired Each basis vector state; and a supplementary set selected from the symmetry space of the chemical system. K supplemental A number of basis states are used to define the remainder of the core space, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system. K supplemental Each of the basis states is not the one described. K selected Members of the basis vector states, and K supplemental + K selected = K desired .

[0296] Terms and Conditions 3 : The method according to any one of clauses 1 to 22, wherein the selected KEach basis vector state also includes: selecting a set of initial base vectors from the proposed space. K selected basis vector states, wherein K selected Each basis vector defines the core space of the chemical system, and its number is greater than the set of basis vectors that define the core space. K desired Each basis vector state; from the set of K selected Choose a set of basis states to remove. K removal There are basis states, among which K selected - K removal = K desired ; and from the above K selected Remove the aforementioned from each of the basis vector states K removal Each basis vector state is used to generate the set of eigenvalues ​​from which the characteristic solution is computed. K desired Each basis vector state.

[0297] Terms and Conditions 4 : The method according to any one of clauses 1 to 22, wherein the selection is based on the electronic conservation selection protocol. K A number of basis states are used to define the core space; wherein when a given basis state from the proposed space is represented by a bit string containing a number of "1"s equal to the number of electrons considered when calculating the characteristic solution of the chemical system, the electron conservation selection protocol selects the given basis state to include in the core space. K In each of the basis vector states.

[0298] Terms and Conditions 5 : The method according to any one of clauses 1 to 22, wherein the selection is based on the α-β electron conservation selection protocol. K A set of basis states is used to define the core space of the chemical system; wherein the α-β electron conservation selection protocol selects the given basis state to be included in the core space when a given basis state from the proposed space is represented by a bit string in which the number of "1"s contained at the corresponding α position is equal to the number of electrons in the α orbital, and the number of "1"s contained at the corresponding β position is equal to the number of electrons in the β orbital considered when calculating the characteristic solution of the chemical system. K In each of the basis vector states.

[0299] Terms and Conditions 6 : The method according to any one of clauses 1 to 22, wherein the selection is based on an overlapping partition selection protocol with partition bias. K A number of basis states are used to define the core space of the chemical system, wherein the overlap partitioning selection protocol selects a given basis state from the proposed space to be included in the... K In each basis state: a graph comprising multiple nodes is constructed, wherein each node corresponds to a basis state of the proposed space; multiple edges are added between the multiple nodes of the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; and all nodes connected to a reference node via one or more edges are selected from the multiple nodes according to a partitioning bias, wherein the partitioning bias omits any nodes among the multiple nodes that cannot be linked to the reference node via one or more edges in the selection.

[0300] Terms and Conditions 7 : The method according to any one of clauses 1 to 22, wherein the selection is based on an overlapping partition selection protocol with a breadth-first bias. K A number of basis states are used to define the core space, wherein the overlapping partition selection protocol selects a given basis state from the proposed space to be included in the core space in such a way as follows: K In each basis state: a graph comprising multiple nodes is constructed, wherein each node corresponds to a basis state in the proposed space; multiple edges are added between the multiple nodes of the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; and nodes are selected from the multiple nodes starting from a reference node according to a breadth-first preference bias, the breadth-first preference biasing to select nodes that are separated from the reference node by fewer edges rather than nodes that are separated from the reference node by more edges, until a node has been selected. K A node or all edges have been traversed.

[0301] Terms and Conditions 8 : The method according to any one of clauses 1 to 22, wherein the selection is based on an overlapping partition selection protocol with optimal priority bias. K A number of basis states are used to define the core space, wherein the overlapping partition selection protocol selects a given basis state from the proposed space to be included in the core space in such a way as follows: KIn each basis state: a graph comprising multiple nodes is constructed, wherein each node corresponds to a basis state in the proposed space; multiple edges are added between the multiple nodes of the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; a heuristic score is assigned to each node; and a node is selected from the multiple nodes starting from a reference node and one or more subsequently selected nodes according to an optimal preference bias, the optimal preference bias selecting nodes connected to the reference node or the subsequently selected node by a single edge and having a higher heuristic score value, rather than nodes having a lower heuristic score value, until a node has been selected. K A node or all edges have been traversed.

[0302] Terms and Conditions 9 : The method according to any one of clauses 1 to 22, wherein the selection is made according to a contribution-based iterative selection protocol from among […]. N Selecting from the hypothetical space of the basis vector states K Each basis vector state is used to define the core space, wherein the contribution-based iterative selection protocol is executed by applying an electron conservation selection protocol or an α-β electron conservation selection protocol to generate a set of... S There are basis states, among which K < S < N From the above S The pair with the highest probability among the basis vector states. K Each basis vector state is experimentally selected to generate a set of... K An experimentally selected basis state and a set of SK The remaining basis states; using the classical computing system K An experimentally selected basis vector constructs a subspace Hamiltonian. H K Hamiltonian from the subspace via the feature solver H k Calculate the characteristic solution of the chemical system; based on the characteristic solution... K The amplitude of each of the experimentally selected basis states α i From the above K Selected from experimentally chosen basis states M There are basis states, among which M < K The amplitude α i Indicated by K The degree of contribution of each of the experimentally selected basis states to the characteristic solution; for the SK Each of the remaining basis states is assigned a significance score, the significance score indicating the significance of the remaining basis states. SK The degree of contribution of each of the remaining basis states to the characteristic solution; from the S Select from the basis states that have the preceding... R The highest significance score R Each basis state, including those not included in the... K At least one basis state from a set of experimentally selected basis states; and the basis states... M each basis vector state and the R A set of basis vectors are combined to produce a set of basis vectors. K The evaluated basis states are used to define the core space.

[0303] Terms and Conditions 10 : The method according to any one of clauses 1 to 22, wherein the selected K Each basis vector state also includes: selecting a set of initial base vectors from the proposed space. K 1 Each basis vector state is used to define the initial core space of the chemical system; in response to determining the selected... K 1 The number of basis states is insufficient for the feature solver to generate the feature solution of the chemical system from the initial core space, thus determining a solution different from the one obtained from the initial core space. K 1 of K 2 Values; and a set of values ​​selected from the proposed space of the chemical system. K 2 each of the basis vector states is selected according to the selection protocol. K 2 Each basis vector state is used to define the second core space of the chemical system.

[0304] Terms and Conditions 11 : The method according to any one of clauses 1 to 22, wherein the selected K Each basis vector state also includes: selecting a set of initial base vectors from the proposed space. K One basis state, wherein the initial selection protocol is used to select the basis state. K One basis vector state is used to define the initial core space of the chemical system; in response to determining the selected... K The number of basis states is insufficient for the characteristic solver to generate characteristic solutions of the chemical system from the initial core space: a set of initial K1 basis states of the chemical system are selected from the proposed space. K 2basis states, wherein the are selected according to a second selection protocol different from the initial selection protocol. K 2 A number of basis vector states are used to define a second core space of the chemical system; the initial core space and the second core space are merged to define the core space as the union of the initial core space and the second core space.

[0305] Terms and Conditions 12 : The method according to any one of clauses 1 to 22, wherein the selected K Each basis vector state also includes: selecting a set of initial vectors from the initial hypothetical space of the chemical system according to the selection protocol. K One basis vector state is used to define the initial core space of the chemical system; in response to determining the selected... K The number of basis states is insufficient for the feature solver to generate the feature solution of the chemical system from the initial core space: a second hypothetical space is generated corresponding to at least one basis state not included in the initial hypothetical space; a set of basis states is selected from the second hypothetical space of the chemical system. K 2 each of the basis vector states is selected according to the selection protocol. K 2 A number of basis vectors are used to define a second core space of the chemical system; and the initial core space and the second core space are merged to define the core space as a union core space.

[0306] Terms and Conditions 13 : The method according to any one of clauses 1 to 22, wherein the selected K Each basis state further includes: selecting from the proposed space according to the selection protocol. K1 A first portion of the core space of the chemical system is defined by basis states; and an additional portion of the core space is defined by exchanging the α and β values ​​of the basis states in the proposed space to create additional basis states outside the proposed space.

[0307] Terms and Conditions 14 : The method according to any one of clauses 1 to 22, wherein the selection is made according to the selection protocol. KA subset of basis states is used to define a first portion of the core space of the chemical system, and the selection further includes: constructing a randomly selected subset of basis states from the symmetric space of the chemical system, wherein the symmetric space comprises fewer basis states than the complete basis space of the chemical system, the basis states not included in the proposed space; calculating the Hamming distance of the randomly selected subset of basis states; constructing a probability distribution based on the Hamming distance of the randomly selected subset of basis states; and sampling from the randomly selected subset of basis states according to the probability distribution. M Each basis vector state is used to define the second part of the core space.

[0308] Terms and Conditions 15 : The method according to any one of clauses 1 to 22, wherein the selected basis vectors are chosen from a list of unsorted basis vector states included in the proposed space. K K basis states are selected based on probability values ​​that are higher than a probability threshold.

[0309] Terms and Conditions 16 : The method according to any one of clauses 1 to 22, wherein: K Each basis vector is generated from the proposed space of the chemical system. m A set of basis states in a set of basis states, wherein m Each set of basis states in the set includes k There are basis states, among which k For the above m The values ​​of each set of basis states in the set of basis states are different; construct m The subspace Hamiltonian, the m Each subspace Hamiltonian in the subspace Hamiltonian corresponds to the... m A set of basis states in the set of basis states; the chemical system is calculated via the feature solver. m Each of the 10 characteristic solutions corresponds to one of the 10 characteristic solutions. m One Hamiltonian among the Hamiltonians of a subspace; obtained from the classical computer system by the aforementioned... m The characteristic solution calculates the weighted sum of the characteristic solutions of the complete Hamiltonian of the chemical system; and outputs the weighted sum of the characteristic solutions of the chemical system.

[0310] Terms and Conditions 17 : The method according to any one of clauses 1 to 22, wherein the selection according to the selection protocol KThe method further includes: defining an initial core space for the chemical system based on basis states included in the initial core space via a classical computer system to identify an extended space, wherein the extended space includes at least one basis state not included in the proposed space; and merging the initial core space and the extended space to define the core space as an extended core space.

[0311] Terms and Conditions 18 : According to any one of Clauses 1 to 22, the method of merging the initial core space and the extended space further includes: compressing only the initial core space, only the extended space, or only the extended core space to include no more than K Each basis vector state.

[0312] Terms and Conditions 19 : The method according to any one of clauses 1 to 22, wherein the chemical system selected from the proposed space of the chemical system K The method further includes: selecting a first set of K basis states from a first hypothetical space to define the core space as the first core space; calculating a first characteristic solution of the chemical system from the first core space via the characteristic solver; in response to determining that the first characteristic solution has not converged: preparing a second state for a test wavefunction of the chemical system, wherein the structure of the quantum circuit for preparing the second state is altered relative to the structure of the quantum circuit for preparing the first state; generating a second hypothetical space from the test wavefunction based on the second state via the quantum computer system, wherein the second hypothetical space comprises fewer basis states than the complete basis space of the chemical system; and selecting a second set of K basis states from the second hypothetical space according to the selection protocol. K A number of basis vector states are used to define a second core space of the chemical system; a second characteristic solution of the chemical system is computed from the second core space via the characteristic solver; and in response to determining that the second characteristic solution has indeed converged, the second characteristic solution of the chemical system is output.

[0313] Terms and Conditions 20 : The method according to any one of clauses 1 to 22, wherein the fixing K The maximum value selection protocol is used as a secondary selection protocol in conjunction with different primary selection protocols.

[0314] Terms and Conditions 21 : The method according to any one of clauses 1 to 22 further includes: simulating the chemical system in a state relative to the target chemical system using the characteristic solution via a classical computer system; and determining the most stable configuration and properties of the chemical system.

[0315] Terms and Conditions 22 : The method according to any one of Clauses 1 to 22, wherein the selection is based on the computing power of the classical computer system. K The value, the computing power is used in K × K Constructing a matrix including K The subspace Hamiltonian of each basis vector state is used to represent the complete Hamiltonian of the chemical system.

[0316] Terms and Conditions 23 : A method for improving computational system efficiency and accuracy in computing Hamiltonian eigenvalues, the method comprising: selecting a first set of basis states from a hypothetical space of a chemical system, wherein: the hypothetical space consists of a subset of basis states from the complete basis space of the chemical system, the basis states being sampleable from a single parameterized quantum circuit on a quantum computer system, the complete basis space consisting of each basis state describing the chemical system; selecting the basis states for the first set to define a first core space of the chemical system according to a first selection protocol; computing a first eigenvalue solution of the chemical system from the first core space via an eigenvalue solver; in response to determining that the first eigenvalue solution has not converged: selecting a second set of basis states to define a second core space of the chemical system; computing a second eigenvalue solution of the chemical system from the core space via the eigenvalue solver; and in response to the second eigenvalue solution converging, outputting the second eigenvalue solution of the chemical system.

[0317] Terms and Conditions 24 : The method according to any one of clauses 23 to 38, wherein the first number of basis states selected for the first group is different in number from the second number of basis states selected for the second group.

[0318] Terms and Conditions 25 : According to any one of Clauses 23 to 38, the first number of basis states selected for the first group is equal in number to the second number of basis states selected for the second group.

[0319] Terms and Conditions 26 : According to any one of Clauses 23 to 38, the first selection protocol is selected from the following: an electron conservation selection protocol; an α-β electron conservation selection protocol; an overlapping partition selection protocol with partition bias; an overlapping partition selection protocol with breadth-first bias; an overlapping partition selection protocol with best-first bias; an iterative selection protocol based on contribution; and a fixed selection protocol. K A maximum value selection protocol; and a protocol for selecting from an unsorted list of basis states included in the proposed space based on probability values ​​that are above a probability threshold.

[0320] Terms and Conditions 27 : The method according to any one of clauses 23 to 38, wherein the second group is selected according to the first selection protocol.

[0321] Terms and Conditions 28 : The method according to any one of Clauses 23 to 38, wherein the second group is selected according to a second selection protocol different from the first selection protocol.

[0322] Terms and Conditions 29 : According to any one of clauses 23 to 38, selecting the first set of ground states further includes: selecting an initial set from the proposed space. K selected Each basis vector state defines a first portion of the first core space of the chemical system, and is fewer in number than a set defining the first core space. K desired Each basis vector state; and a supplementary set selected from the symmetry space of the chemical system. K supplemental Several basis states are used to define the remainder of the first core space, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system, and wherein each of the supplementary basis states is not one of the... K selected Members of the basis vector states, and K supplemental + K selected = K desired .

[0323] Terms and Conditions 30 : According to any one of clauses 23 to 38, selecting the first set of ground states further includes: selecting an initial set from the proposed space. K selected basis vector states, wherein K selectedEach basis vector defines the first core space of the chemical system, and its number is greater than the set of base vectors that define the core space. K desired Each basis vector state; from the set of K selected Choose a set of basis states to remove. K removal There are basis states, among which K selected - K removal = K desired ; and from the above K selected Remove the aforementioned from each of the basis vector states K removal Each basis vector state is used to generate the set of basis vector states from which the first characteristic solution is computed. K desired Each basis vector state.

[0324] Terms and Conditions 31 : According to any one of clauses 23 to 38, the method of selecting the second set of ground states further includes: generating a second hypothetical space, the second hypothetical space being different from the hypothetical space from which the first set of basis states was selected.

[0325] Terms and Conditions 32 : The method according to any one of clauses 23 to 38, wherein the second set of basis states is selected from the proposed space.

[0326] Terms and Conditions 33 : The method according to any one of clauses 23 to 38, wherein: the first set of basis states and the second set of basis states are generated from the proposed space of the chemical system. m Two sets of basis states in a set of basis states, wherein m Each set of basis states in the set includes k There are basis states, among which k For the above m The values ​​of each set of basis states in the set of basis states are different; construct m The subspace Hamiltonian, the m Each subspace Hamiltonian in the subspace Hamiltonian corresponds to the... m A set of basis states in the set of basis states; the chemical system is calculated via the feature solver. m Each of the 10 characteristic solutions corresponds to one of the 10 characteristic solutions. m One of the Hamiltonians in the subspace Hamiltonians, the first characteristic solution and the second characteristic solution are the Hamiltonians of the subspace Hamiltonians.m Two characteristic solutions from a set of characteristic solutions; obtained from the above using a classical computer system. m The characteristic solution calculates the weighted sum of the characteristic solutions of the complete Hamiltonian of the chemical system; and outputs the weighted sum of the characteristic solutions of the chemical system.

[0327] Terms and Conditions 34 : The method according to any one of clauses 23 to 38 further comprises: expanding the first core space via a classical computer system based on basis states included in the first core space to identify an expanded space, wherein the expanded space includes at least one basis state not included in the proposed space; and merging the first core space and the expanded space to define the first core space as the expanded core space.

[0328] Terms and Conditions 35 : According to any one of clauses 23 to 38, the method further includes: selecting from the proposed space. K 1 A first portion of the first core space of the chemical system is defined by a basis state; additional basis states outside the proposed space are created by exchanging the α and β values ​​representing the basis states in the proposed space to define a second portion of the core space; and when the first characteristic solution is computed, the first portion and the second portion of the first core space are included.

[0329] Terms and Conditions 36 : According to any one of clauses 23 to 38, the method further includes: selecting from the proposed space. K A subset of basis states is defined to define a first part of the core space of the chemical system; a randomly selected subset of basis states is constructed from the symmetric space of the chemical system, wherein the symmetric space comprises fewer basis states than the complete basis space of the chemical system, the basis states not included in the proposed space; Hamming distances of the randomly selected subset of basis states are calculated; a probability distribution is constructed based on the Hamming distances of the randomly selected subset of basis states; and samples are taken from the randomly selected subset of basis states according to the probability distribution. M A number of basis vectors are used to define a second part of the core space; and when the first feature solution is computed, it includes the first part and the second part of the first core space.

[0330] Terms and Conditions 37 : The method according to any one of clauses 23 to 38 further comprises: simulating the chemical system in a state relative to the target chemical system using the second characteristic solution via a classical computer system; and determining the most stable configuration and properties of the chemical system.

[0331] Terms and Conditions 38 : According to any one of clauses 23 to 38, the number of basis states in the first group is K1, wherein the value of K1 is selected based on the computing power of a classical computer system, the computing power being used to construct a subspace Hamiltonian comprising K1 basis states in a K1×K1 matrix to represent the full Hamiltonian of the chemical system.

[0332] Terms and Conditions 39 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting a first set of... K A set of basis states, wherein the hypothetical space consists of a subset of basis states from the complete basis space of the chemical system, the basis states being sampled from a single parameterized quantum circuit on a quantum computer system, wherein the complete basis space consists of each basis state describing the chemical system, wherein the basis states are selected according to a selection protocol. K A number of basis vectors are used to define the core space of the chemical system; characteristic solutions of the chemical system are calculated from the core space via a characteristic solver; and the characteristic solutions of the chemical system are output.

[0333] Terms and Conditions 40 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting an initial set of... from the hypothetical space of the chemical system... K selected A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K selected A number of basis vector states are used to define the first part of the core space of the chemical system, and are greater in number than a set of basis vector states used to define the core space to compute the characteristic solutions of the chemical system via the characteristic solver. K desired One basis vector state; a supplementary set is selected from the symmetry space of the chemical system. K supplemental A number of basis states are used to define the remainder of the core space, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system. Ksupplemental Each of the basis states is not the one described. K selected Members of the basis vector states, and K supplemental + Kselected = K desired The characteristic solution of the chemical system is calculated from the core space via the characteristic solver; and the characteristic solution of the chemical system is output.

[0334] Terms and Conditions 41 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting an initial set of K from the hypothetical space of the chemical system. selected A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K selected A number of basis vector states are used to define the first part of the core space of the chemical system, and are greater in number than a set of basis vector states used to define the core space to compute the characteristic solutions of the chemical system via the characteristic solver. K desired Each basis vector state; from the set of K selected Choose a set of basis states to remove. K removal There are basis states, among which K selected - K removal = K desired From the above K selected Remove the aforementioned from each of the basis vector states K removal Each basis vector state is used to generate the set of eigenvalues ​​from which the characteristic solution is computed. K desired The system comprises: a basis vector state; a characteristic solution of the chemical system is calculated from the core space via the characteristic solver; and the characteristic solution of the chemical system is output.

[0335] Terms and Conditions 42 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < NThe selection is based on the electronic conservation selection protocol. K A number of basis vector states are used to define the core space of the chemical system; wherein when the origin is from the N When a given basis state is represented by a bit string containing a number of "1"s equal to the number of electrons considered in calculating the characteristic solution of the chemical system, the electron conservation selection protocol selects the given basis state to include the given basis state in the given basis state. K In each basis vector state; the characteristic solution of the chemical system is calculated from the core space via a characteristic solver; and the characteristic solution of the chemical system is output.

[0336] Terms and Conditions 43 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on the α-β electron conservation selection protocol. K Each basis vector state is used to define the core space of the chemical system, wherein when the vector state originates from the... N When a given basis state is represented by a bit string containing the number of "1"s at the corresponding α positions equal to the number of electrons in the α orbitals, and the number of "1"s at the corresponding β positions equal to the number of electrons in the β orbitals considered when calculating the characteristic solutions of the chemical system, the α-β electron conservation selection protocol selects the given basis state to include in the given... K In each basis vector state; the characteristic solution of the chemical system is calculated from the core space via a characteristic solver; and the characteristic solution of the chemical system is output.

[0337] Terms and Conditions 44 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on an overlapping partition selection protocol with partition bias. K A number of basis vector states are used to define the core space of the chemical system, wherein the overlapping partition selection protocol is derived from the following... N A given basis state is selected from the basis states to include the basis states in the... KIn each basis vector state: construct a graph comprising multiple nodes, where each node corresponds to the... N One of the basis states; adding multiple edges between the plurality of nodes in the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1> and |b2> where <b1|b2> is nonzero; selecting all nodes from the plurality of nodes connected to the reference node via one or more edges according to a partitioning bias, the partitioning bias omitting any nodes from the plurality of nodes that cannot be linked to the reference node via one or more edges in the selection; calculating the characteristic solution of the chemical system from the core space via a characteristic solver; and outputting the characteristic solution of the chemical system.

[0338] Terms and Conditions 45 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on an overlapping partitioning selection protocol with a breadth-first bias. K A number of basis vector states are used to define the core space of the chemical system, wherein the overlapping partition selection protocol is derived from the following... N A given basis state is selected from the basis states to include the basis states in the... K In each basis vector state: construct a graph comprising multiple nodes, where each node corresponds to the... N One of the basis states; adding multiple edges between the plurality of nodes in the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; and selecting nodes from the plurality of nodes starting from the reference node according to a breadth-first preference bias, the breadth-first preference biasing to select nodes that are fewer edges away from the reference node rather than nodes that are more edges away from the reference node, until a node has been selected. K The system has traversed all nodes or edges; computed the characteristic solution of the chemical system from the core space via the characteristic solver; and output the characteristic solution of the chemical system.

[0339] Terms and Conditions 46 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on an overlapping partition selection protocol with the best priority bias. K A number of basis vector states are used to define the core space of the chemical system, wherein the overlapping partition selection protocol is derived from the following... N A given basis state is selected from the basis states to include the basis states in the... K In the N basis states: a graph comprising multiple nodes is constructed, where each node corresponds to one of the N basis states; multiple edges are added between the multiple nodes of the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is non-zero; a heuristic score is assigned to each node; and a node is selected from the multiple nodes starting from a reference node and one or more subsequently selected nodes according to a best priority bias, the best priority bias selecting nodes connected to the reference node or the subsequently selected node by a single edge and having a higher heuristic score value, rather than nodes having a lower heuristic score value, until a node has been selected. K The system has traversed all nodes or edges; computed the characteristic solution of the chemical system from the core space via the characteristic solver; and output the characteristic solution of the chemical system.

[0340] Terms and Conditions 47 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on a contribution-based iterative selection protocol. K Each basis vector state defines the core space of the chemical system, wherein the contribution-based iterative selection protocol is executed by applying an electron conservation selection protocol or an α-β electron conservation selection protocol to generate a set of basis vector states that satisfy the electron conservation selection protocol or the α-β electron conservation selection protocol. S There are basis states, among which K < S < N From the above S The pair with the highest probability among the basis vector states. K Each basis vector state is experimentally selected to generate a set of... K An experimentally selected basis state and a set of SKThe remaining basis states; using the classical computing system K An experimentally selected basis vector constructs a subspace Hamiltonian. H K Hamiltonian from the subspace via the feature solver H K Calculate the characteristic solution of the chemical system; based on the characteristic solution... K The amplitude of each of the experimentally selected basis states α i From the above K Selected from experimentally chosen basis states M There are basis states, among which M < K The amplitude α i Indicated by K The degree of contribution of each of the experimentally selected basis states to the characteristic solution; for the SK Each of the remaining basis states is assigned a significance score, the significance score indicating the significance of the remaining basis states. SK The degree of contribution of each of the remaining basis states to the characteristic solution; from the S Select from the basis states that have the preceding... R The highest significance score R Each basis state, including those not included in the... K At least one basis state from a set of experimentally selected basis states; M each basis vector state and the R A set of basis vectors are combined to produce a set of basis vectors. K The evaluated basis states are used to define the core space; the characteristic solution of the chemical system is calculated from the core space via a characteristic solver; and the characteristic solution of the chemical system is output.

[0341] Terms and Conditions 48 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N , among which according to fixed K A maximum value selection protocol is used to select the... K A number of basis vector states are used to define the core space of the chemical system, wherein the fixed vector states are... K The maximum value selection protocol is based on the N The probability of each basis state is related to the NSort the basis vector states and from the... N The K basis states with the highest probability are selected from the basis states to define the core space; the characteristic solution of the chemical system is calculated from the core space via a characteristic solver; and the characteristic solution of the chemical system is output.

[0342] Terms and Conditions 49 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting an initial set of... from the hypothetical space of the chemical system... K 1 A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K 1 Each basis vector state is used to define the initial core space of the chemical system; in response to determining the selected... K 1 The number of basis states is insufficient for the feature solver to generate the feature solution of the chemical system from the core space, thus determining a solution different from the one obtained from the core space. K 1 of K 2 Values; select a set from the proposed space of the chemical system. K 2 each of the basis vector states is selected according to the selection protocol. K 2 A number of basis vector states are used to define a second core space of the chemical system; a characteristic solution of the chemical system is calculated from the second core space via a characteristic solver provided by at least one of the classical computer system, the quantum computer system, or the hybrid quantum-classical computer system; and the characteristic solution of the chemical system is output.

[0343] Terms and Conditions 50 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting an initial set of... from the hypothetical space of the chemical system... K One basis state, wherein the proposed space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis state is selected according to an initial selection protocol. K 1 Each basis vector state is used to define the initial core space of the chemical system; in response to determining the selected... KThe number of basis states is insufficient for the characteristic solver to generate characteristic solutions of the chemical system from the initial core space: a set of initial K1 basis states of the chemical system are selected from the proposed space. K 2 basis states, wherein the are selected according to a second selection protocol different from the initial selection protocol. K 2 The initial core space and the second core space are combined to define a union core space; the characteristic solution of the chemical system is calculated from the union core space via a characteristic solver; and the characteristic solution of the chemical system is output.

[0344] Terms and Conditions 51 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting an initial set of initial hypothetical solutions from the initial hypothetical space of the chemical system. K One basis state, wherein the initial hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis state is selected according to a selection protocol. K 1 Each basis vector state is used to define the initial core space of the chemical system; in response to determining the selected... K The number of basis states is insufficient for the feature solver to generate a feature solution for the chemical system from the initial core space: a second hypothetical space is generated corresponding to at least one basis state not included in the initial hypothetical space; a set of basis states is selected from the second hypothetical space of the chemical system. K 2 each of the basis vector states is selected according to the selection protocol. K 2 The initial core space and the second core space are combined to define a union core space; the characteristic solution of the chemical system is calculated from the union core space via a characteristic solver; and the characteristic solution of the chemical system is output.

[0345] Terms and Conditions 52 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K 1 A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K 1A first portion of the core space of the chemical system is defined by using basis states; additional basis states outside the proposed space are created by exchanging the α and β values ​​representing the basis states in the proposed space to define a second portion of the core space; a characteristic solution of the chemical system is computed from the core space including the first and second portions via a characteristic solver; and the characteristic solution of the chemical system is output.

[0346] Terms and Conditions 53 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K A subset of basis states is defined to define a first part of the core space of the chemical system; a randomly selected subset of basis states is constructed from the symmetric space of the chemical system, wherein the symmetric space comprises fewer basis states than the complete basis space of the chemical system, and the basis states are not included in the proposed space; Hamming distances of the randomly selected subset of basis states are calculated; a probability distribution is constructed based on the Hamming distances of the randomly selected subset of basis states; and samples are taken from the randomly selected subset of basis states according to the probability distribution. M A number of basis vectors are used to define a second part of the core space; a characteristic solution of the chemical system is calculated from the core space including the first part and the second part via a characteristic solver; and the characteristic solution of the chemical system is output.

[0347] Terms and Conditions 54 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected from an unsorted list of basis states included in the hypothetical space based on probability values ​​having a probability threshold. K A number of basis vector states; calculate the characteristic solution of the chemical system from the core space via a characteristic solver; and output the characteristic solution of the chemical system.

[0348] Terms and Conditions 55 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: generating from the hypothetical space of a chemical system m The basis vector states, whereinm Each set of basis states in the set includes k There are basis states, among which k The values ​​of each of the m sets of basis states are different; construct... m The subspace Hamiltonian, the m Each subspace Hamiltonian in the subspace Hamiltonian corresponds to the... m A set of basis states in the basis vector state set; the chemical system is calculated via a characteristic solver. m Each of the 10 characteristic solutions corresponds to one of the 10 characteristic solutions. m One Hamiltonian among the Hamiltonians of a subspace; obtained from the classical computer system by the aforementioned... m The characteristic solutions are calculated by weighting and summing the characteristic solutions of the complete Hamiltonian of the chemical system; and the characteristic solutions of the chemical system are output.

[0349] Terms and Conditions 56 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting from the hypothetical space of the chemical system K A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K A set of basis states are used to define an initial core space for the chemical system; the initial core space is expanded via a classical computer system based on the basis states included in the initial core space to identify an expanded space, wherein the expanded space includes at least one basis state not included in the proposed space; the initial core space and the expanded space are merged to define an expanded core space; a characteristic solution of the chemical system is computed from the expanded core space via a characteristic solver; and the characteristic solution of the chemical system is output.

[0350] Terms and Conditions 57 : According to the method described in Clause 56, merging the initial core space and the extended space further includes: compressing only the initial core space, only the extended space, or only the extended core space to include no more than K Each basis vector state.

[0351] Terms and Conditions 58 : A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: selecting a first set of... from a first hypothetical space of the chemical system. KA number of basis states, wherein the first hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the first state is selected from the first hypothetical space according to a selection protocol. K A first characteristic solution of the chemical system is calculated from the first core space via a characteristic solver; in response to determining that the first characteristic solution has not converged: a second state is prepared for the test wavefunction, wherein the structure of the quantum circuit used to prepare the second state is altered relative to the structure used to prepare the first state; a second hypothetical space is generated from the test wavefunction via the quantum computer system based on the second state, wherein the second hypothetical space comprises fewer basis states than the complete basis space of the chemical system; a second set of basis states is selected from the second hypothetical space according to the selection protocol. K A number of basis vector states are used to define a second core space of the chemical system; a second characteristic solution of the chemical system is computed from the second core space via the characteristic solver; and in response to determining that the second characteristic solution has indeed converged, the second characteristic solution of the chemical system is output.

[0352] Terms and Conditions 59 : A method for improving computational system efficiency and accuracy in computing characteristic solutions of a Hamiltonian, the method comprising: creating a complete Hamiltonian having a Hilbert space with 2n basis states via a classical computer system; running a first plurality of iterations on a hybrid quantum-classical computer system including the classical computer system and a quantum computer system until a first stopping criterion is met, wherein each iteration of the first plurality of iterations comprises: assigning an initial state to the complete Hamiltonian via the classical computer system via an initial state preparation protocol; generating a test wavefunction based on the initial state via the hybrid quantum-classical computer system using a test wavefunction preparation protocol; and, on the hybrid quantum-classical computer system, performing a first plurality of iterations on the test wavefunction based on the initial state ... A second series of iterations is run on the hybrid quantum-classical computer system until a second stopping criterion is met, wherein each iteration of the second series of iterations includes: sampling a batch of results including values ​​from the test wavefunction via the quantum computer system, wherein: the values ​​represent computational basis states of the Hilbert space; stopping the second series of iterations on the hybrid quantum-classical computer system in response to the satisfaction of the second stopping criterion, wherein a first plurality of values ​​are output, the first plurality of values ​​including the batch of results sampled in each iteration of the second series of iterations; and selecting a second plurality of values ​​from the first plurality of values ​​that satisfy the criterion via the classical computer system, the selection being... The selection of values ​​from the first plurality of values ​​is performed according to a selection protocol, wherein: the number of values ​​in the second plurality of values ​​is equal to K; K is less than 2n; and the K values ​​in the second plurality of values ​​describe a subset of the K basis states with the largest magnitude among the 2n basis states; the selection protocol includes one of the following: a symmetry criterion; an overlap criterion; and a contribution-based iteration criterion; the second plurality of values ​​are used via the classical computer system to construct a subspace Hamiltonian including K basis states in the subspace Hamiltonian representation to represent the complete Hamiltonian; the minimum feature of the subspace Hamiltonian is computed via a feature solver provided by the hybrid quantum-classical computer system. The process involves: calculating eigenvalues; computed an eigenvector corresponding to the minimum eigenvalue via the eigenvalue solver provided by the hybrid quantum-classical computer system; computed, via the classical computer system, an approximate ground state value representing the ground state of the complete Hamiltonian and an approximate ground state energy value representing the ground state energy of the complete Hamiltonian using the minimum eigenvalue of the subspace Hamiltonian and the eigenvector corresponding to the minimum eigenvalue of the subspace Hamiltonian; stopping the first multiple iterations on the classical computer system in response to satisfying the first stopping criterion; and outputting the ground state value and the ground state energy value from the final iteration of the first multiple iterations.

[0353] Terms and Conditions 60 : According to any one of Clauses 59 to 80, the initial state preparation protocol includes: assigning the initial state to the complete Hamiltonian via the classical computer system, wherein the initial state includes at least one of: a state prepared via the Hartree-Fock protocol; a zero state; a computational basis state of the Hilbert space; a state prepared via an ab initio initial state preparation protocol; a state prepared via a tensor network initial state preparation protocol; a sparse initial state generated from the ground state value of the final iteration of a previous first multiple iteration; a uniformly distributed state; and a randomly distributed state.

[0354] Terms and Conditions 61 : According to any one of clauses 59 to 80, the experimental wavefunction preparation protocol comprises: running a third multiple iteration via a variational quantum characteristic solver (VQE) on the hybrid quantum-classical computer system until a third stopping criterion is met, wherein each iteration in the third multiple iteration comprises: constructing a parameterized quantum circuit based on the initial state via the classical computer system; generating a parameterized wavefunction approximating the ground state of the complete Hamiltonian via executing the parameterized quantum circuit on the quantum computer system; performing Pauli sampling on the parameterized wavefunction generated by the parameterized quantum circuit with respect to the complete Hamiltonian via the quantum computer system; calculating the expectation value of the parameterized wavefunction relative to the complete Hamiltonian via the classical computer system; and updating a set of parameters of the parameterized quantum circuit based on the expectation value; stopping the third multiple iteration on the hybrid quantum-classical computer system in response to meeting the third stopping criterion, wherein the parameterized wavefunction from the last iteration of the third multiple iteration is output; and wherein the third multiple iteration is performed prior to the second multiple iteration.

[0355] Terms and Conditions 62 : The method according to any one of clauses 59 to 80, wherein the third stopping criterion is satisfied in response to at least one of the following: a predefined number of iterations in the third multiple iterations have been performed; the third multiple iterations have been run for a predefined amount of time; the quantum computer system has been used for a predefined amount of time; the change of the expected value from a given iteration in the third multiple iterations to subsequent iterations in the third multiple iterations is within a threshold of the quantum noise change value of the quantum computer system; the first derivative of the expected value from a given iteration in the third multiple iterations falls below a termination threshold; the change of the expected value from a plurality of previous iterations in the third multiple iterations to subsequent iterations in the third multiple iterations is within a termination threshold; and a predefined number of samples across the third multiple iterations, wherein the samples include: obtaining a wavefunction by executing quantum circuitry via the quantum computer system; obtaining a measurement result by performing a measurement on the wavefunction via the quantum computer system; and outputting the measurement result.

[0356] Terms and Conditions 63 : The method according to any one of clauses 59 to 80, wherein: the experimental wavefunction preparation protocol comprises: constructing a parameterized quantum circuit based on the initial state via the classical computer system; generating a parameterized wavefunction approximating the ground state of the complete Hamiltonian via executing the parameterized quantum circuit on the quantum computer system without performing Pauli sampling; and outputting the parameterized wavefunction.

[0357] Terms and Conditions 64 : The method according to any one of clauses 59 to 80, wherein the second stopping criterion is satisfied in response to at least one of the following: a predefined number of iterations in the second multiple iterations has been performed; and the batch result contains a predefined number of values.

[0358] Terms and Conditions 65 : The method according to any one of clauses 59 to 80, wherein the inclusion criterion is satisfied in response to at least one of the following: the second plurality of values ​​contains a predefined number of values ​​from the first plurality of values; and the number of values ​​contained in the second plurality of values ​​is within a predefined threshold of the predefined number of values ​​from the first plurality of values.

[0359] Terms and Conditions 66 : According to any one of the provisions 59 to 80, the selection protocol includes: when a screening criterion results in not exactly K values ​​being selected from the first plurality of values, selecting an additional screening criterion to select an exact K values ​​for constructing the subspace Hamiltonian, wherein the additional screening criterion is selected from at least one of: a second symmetry criterion; a second screening criterion; an overlap criterion; a contribution-based iterative criterion; randomized selection; a screening criterion; or selection from a third plurality of values ​​from a previous second plurality of iterations.

[0360] Terms and Conditions 67 : The method according to any one of clauses 59 to 80, wherein the symmetry criterion is satisfied in response to the value representing a calculated basis state of the Hilbert space having a predetermined number of electrons, or wherein the screening criterion is satisfied in response to the value representing a calculated basis state of the Hilbert space and the amplitude value being within a predefined highest percentage of a plurality of amplitude values ​​of each of the first plurality of values.

[0361] Terms and Conditions 68 : The method according to any one of clauses 59 to 80, wherein the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K, and the values ​​are represented by a bit string having a number of "1"s equal to the number of electrons in the chemical system represented by the complete Hamiltonian.

[0362] Terms and Conditions 69 : According to any one of the provisions 59 to 80, the method wherein the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K, the values ​​being represented by a bit string having a number of "1"s at positions corresponding to a given orbital equal to the number of electrons in the given orbital of the chemical system represented by the complete Hamiltonian.

[0363] Terms and Conditions 70 : The method according to any one of clauses 59 to 80, wherein the overlap criterion is satisfied based on: generating a graph of a plurality of nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; identifying a reference node associated with a known reference state from the plurality of nodes; and selecting all nodes from the plurality of nodes that share a partition with the reference node.

[0364] Terms and Conditions 71 : The method according to any one of clauses 59 to 80, wherein the overlap criterion is satisfied based on: generating a graph of a plurality of nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; identifying a reference node associated with a known reference state from the plurality of nodes; and selecting all nodes from the plurality of nodes that are a distance from the reference node or a previously selected node by an edge, until all nodes from the plurality of nodes that share a partition with the reference node are selected, or until at least K nodes are selected from the plurality of nodes, whichever occurs first.

[0365] Terms and Conditions 72 : The method according to any one of clauses 59 to 80, wherein the overlap criterion is satisfied based on: generating a graph of a plurality of nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; assigning a score to each of the plurality of nodes based on a heuristic measure of the importance of the corresponding basis vector states; identifying a reference node from the plurality of nodes that is associated with a known reference state; and selecting a next node from the plurality of nodes that is an edge distance from the reference node or a previously selected node, the next node having the highest score relative to all other nodes from the plurality of nodes that are an edge distance from the reference node or any previously selected node, until all nodes from the plurality of nodes that share a partition with the reference node have been selected, or until at least K nodes from the plurality of nodes have been selected, whichever occurs first.

[0366] Terms and Conditions 73 : The method according to any one of clauses 59 to 80, wherein the screening criteria are satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K, and the values ​​exceed a predefined threshold.

[0367] Terms and Conditions 74 : According to any one of clauses 59 to 80, the method wherein the contribution-based iteration criterion is satisfied based on the following: applying a symmetry criterion to the first plurality of values ​​via the classical computing system to obtain a fourth plurality of values; sorting the fourth plurality of values ​​based on a first plurality of magnitude squared values ​​via the classical computing system; selecting a fifth plurality of values ​​from the fourth plurality of values ​​via the classical computing system, wherein: the number of values ​​of the fifth plurality of values ​​is equal to k; constructing a subspace Hamiltonian using the fifth plurality of values ​​via the classical computing system; solving for the eigenvalues ​​of the subspace Hamiltonian constructed using the fifth plurality of values ​​via the classical computing system; selecting a sixth plurality of values ​​based on a second inclusion criterion via the classical computing system, wherein the second inclusion criterion is satisfied based on the following: via the classical computing system... The classical computing system selects a seventh plurality of values ​​with the largest amplitude from the fifth plurality of values; assigns a first plurality of significance scores to values ​​in the fourth plurality of values ​​that are not included in the fifth plurality of values ​​via the classical computing system, wherein the significance scores in the first plurality of significance scores include: a measure of the effect of adding the corresponding value to the characteristic solution of the fifth plurality of values ​​via the classical computing system; selects an eighth plurality of values ​​via the classical computing system, wherein the eighth plurality of values ​​includes: the number of values ​​of the eighth plurality of values ​​is equal to R; and the values ​​in the eighth plurality of values ​​have a significance score within the first R significance scores; combines a ninth plurality of values ​​via the classical computing system, wherein the ninth plurality of values ​​includes: the sixth plurality of values; and the eighth plurality of values; and outputs the ninth plurality of values ​​via the classical computing system.

[0368] Terms and Conditions 75 : According to any one of Clauses 59 to 80, the first stopping criterion is satisfied in response to at least one of the following: a predefined number of iterations in the first plurality of iterations has been performed; the first plurality of iterations have been run for a predefined amount of time; the quantum computer system has been used for a predefined amount of time; the change in the ground state energy value from a given iteration in the first plurality of iterations to subsequent iterations in the first plurality of iterations is within a threshold value of the quantum noise change value of the quantum computer system; the first derivative of the ground state energy value from a given iteration in the first plurality of iterations falls below a termination threshold; and the change in the ground state energy value from a plurality of previous iterations in the first plurality of iterations to subsequent iterations in the first plurality of iterations is within a termination threshold.

[0369] Terms and Conditions 76 : The method according to any one of clauses 59 to 80 further includes, before outputting from the final iteration of the second plurality of iterations the first plurality of values ​​of the batch results included in each iteration of the second plurality of iterations: in response to determining that the second stopping criterion is not met: sampling, via the quantum computer system, a second batch of results including a second value from the test wavefunction in the computational basis, wherein: the second value represents a computational basis state of the Hilbert space.

[0370] Terms and Conditions 77 : The method according to any one of clauses 59 to 80 further includes, before outputting the parameterized wavefunction from the last iteration of the third multiple iterations: in response to determining that the third stopping criterion is not met: constructing a second parameterized quantum circuit according to the initial state via the classical computer system; generating a second parameterized wavefunction approximating the ground state of the complete Hamiltonian via executing the second parameterized quantum circuit on the quantum computer system; performing Pauli sampling on the second parameterized wavefunction generated by the second parameterized quantum circuit with respect to the complete Hamiltonian via the quantum computer system; calculating a second expectation value of the second parameterized wavefunction relative to the complete Hamiltonian via the classical computer system; and updating a second set of parameters of the second parameterized quantum circuit according to the second expectation value.

[0371] Terms and Conditions 78 : The method according to any one of clauses 59 to 80 further includes, before outputting the ground state value and the ground state energy value from the final iteration of the first multiple iterations: in response to determining that the first stopping criterion is not met: assigning a second initial state to the complete Hamiltonian via the initial state preparation protocol via the classical computer system; and rerunning the first multiple iterations on the hybrid quantum-classical computer system until the first stopping criterion is met for the second time.

[0372] Terms and Conditions 79 : The method according to any one of clauses 59 to 80 further includes, before outputting the ground state value and the ground state energy value from the final iteration of the first plurality of iterations: in response to determining that the first stopping criterion is not met: rerunning the second plurality of iterations on the quantum computer system until the second stopping criterion is met for the second time.

[0373] Terms and Conditions 80 : The method according to any one of clauses 59 to 80 further includes, before outputting the ground state value and the ground state energy value from the final iteration of the first plurality of iterations: in response to determining that the first stopping criterion is not met: via the classical computer system, reselecting new second plurality of values ​​from the first plurality of values ​​that satisfy the inclusion criterion, the reselection being performed by selecting values ​​from the first plurality of values ​​according to the selection protocol.

[0374] Terms and Conditions 81 : A method for improving computational system efficiency and accuracy in computing Hamiltonian eigenvalues, the method comprising: creating a complete Hamiltonian having a Hilbert space with 2n basis states via a classical computer system; assigning an initial state to the complete Hamiltonian via the classical computer system; constructing a parameterized quantum circuit based on the initial state via the classical computer system; running multiple iterations of a variational quantum eigenvalue solver (VQE) on a quantum computer system using the parameterized quantum circuit to generate a parameterized wavefunction approximating the ground state of the complete Hamiltonian, wherein each iteration of the multiple iterations comprises: performing Pauli sampling or Pauli measurement on the quantum state generated by the parameterized quantum circuit with respect to the complete Hamiltonian to calculate an expectation value of the parameterized wavefunction relative to the complete Hamiltonian; and updating the parameters of the parameterized wavefunction based on the expectation value; stopping the multiple iterations on the quantum computer system in response to satisfying a stopping criterion, wherein the parameterized wavefunction from the last iteration of the multiple iterations is output as a finalized wavefunction; and resolving the problem from the initial state via the classical computer system. The final wavefunction selects multiple values ​​that satisfy the following criteria: the number of samples in the multiple samples is equal to K; K is less than 2n; the K values ​​in the multiple samples describe a subset of the K basis states with the largest magnitude from the 2n basis states as an approximate representation of the energy states in the complete Hamiltonian; in response to determining that K is within the computational capability of the classical computer system, the computational capability is used to construct a subspace Hamiltonian including K basis states in a K×K matrix to represent the complete Hamiltonian: the K values ​​are calculated via the classical computer system. The classical values ​​of the K basis states are obtained; the subspace Hamiltonian is constructed using the classical values ​​of the K basis states via the classical computer system; the minimum eigenvalue of the subspace Hamiltonian is calculated via the eigenvalue solver provided by the classical computer system; the eigenvector corresponding to the minimum eigenvalue is calculated via the classical computer system; and the ground state value representing the ground state and the ground state energy representing the ground state energy of the complete Hamiltonian are calculated via the classical computer system using the minimum eigenvalue and the eigenvector of the subspace Hamiltonian.

[0375] Terms and Conditions 82 : The method according to any one of clauses 81 to 94 further comprises: simulating a chemical system in a state relative to a target biomolecule via the classical computer system using the ground state value and the ground state energy value, the chemical system being representable as a 2n×2n matrix and corresponding to the complete Hamiltonian; and administering a therapeutically effective dose of the chemical system to a biological subject according to the simulation to treat the condition.

[0376] Terms and Conditions 83 : According to any one of the provisions 81 to 94, the stopping criterion is satisfied in response to one of the following: the change of the expected value from a given iteration in the multiple iterations to a subsequent iteration in the multiple iterations is within a threshold of the quantum noise change value of the quantum computer system; or the first derivative of the expected value across the multiple iterations falls below a termination threshold.

[0377] Terms and Conditions 84 : The method according to any one of clauses 81 to 94 further includes, before determining K within the computational capability range of the classical computer system: determining that an earlier K value determined after satisfying the stopping criterion for earlier iterations is less than the computational capability of the classical computer system; assigning a second initial state to the complete Hamiltonian based on the classical ground state and the classical ground state energy of the complete Hamiltonian via the classical computer system; constructing a second parameterized quantum circuit based on the second initial state via the classical computer system; running a second series of iterations of the VQE on the quantum computer system using the second parameterized quantum circuit to generate a second parameterized wavefunction approximating the ground state of the complete Hamiltonian until the stopping criterion is satisfied; and wherein the plurality of values ​​satisfying the criterion are reselected from the second parameterized wavefunction via the classical computer system, such that K is greater than the earlier K value.

[0378] Terms and Conditions 85 : The method according to any one of clauses 81 to 94 further includes, before determining K within the computational capability range of the classical computing system: determining that an earlier K value determined after satisfying the stopping criterion for earlier iterations is greater than the computational capability of the classical computer system; assigning a second initial state to the complete Hamiltonian based on the classical ground state and the classical ground state energy of the complete Hamiltonian via the classical computer system; constructing a second parameterized quantum circuit based on the second initial state via the classical computer system; running a second series of iterations of the VQE on the quantum computer system using the second parameterized quantum circuit to generate a second parameterized wavefunction approximating the ground state of the complete Hamiltonian until the stopping criterion is satisfied; and wherein the plurality of values ​​satisfying the criterion are reselected from the second parameterized wavefunction via the classical computer system, such that K is less than the earlier K value.

[0379] Terms and Conditions 86 : The method according to any one of clauses 81 to 94, wherein the inclusion standard specifies that the plurality of values ​​are composed of the 2n basis vector states with non-zero values ​​identified by the quantum computing system.

[0380] Terms and Conditions 87 : According to any one of the provisions 81 to 94, the inclusion criterion is based on a selection protocol, the selection protocol comprising: when a screening criterion results in not exactly K values ​​being selected from the first plurality of values, selecting an additional screening criterion to select an exact K values ​​for constructing the subspace Hamiltonian, wherein the additional screening criterion is selected from at least one of: a second symmetry criterion; a second screening criterion; an overlap criterion; randomization selection; a screening criterion; or selection from a third plurality of values ​​from a previous second plurality of iterations.

[0381] Terms and Conditions 88 : The method according to any one of clauses 81 to 94, wherein the including criteria include a symmetry criterion, wherein the symmetry criterion is satisfied in response to the value representing a calculated basis state of the Hilbert space having a predetermined number of electrons, or wherein the screening criterion is satisfied in response to the value representing a calculated basis state of the Hilbert space and the amplitude value being within a predefined highest percentage of a plurality of amplitude values ​​of the first plurality of values.

[0382] Terms and Conditions 89 : According to any one of the provisions 81 to 94, the method wherein the including standard includes a symmetry standard, wherein the symmetry standard is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K, and the values ​​are represented by a bit string having a number of "1"s equal to the number of electrons in the chemical system represented by the complete Hamiltonian.

[0383] Terms and Conditions 90 : According to any one of the provisions 81 to 94, the method wherein the including criterion includes a symmetry criterion, wherein the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K, the values ​​being represented by a bit string having a number of "1"s at positions corresponding to a given orbital equal to the number of electrons in the given orbital of the chemical system represented by the complete Hamiltonian.

[0384] Terms and Conditions 91 : The method according to any one of clauses 81 to 94, wherein the including criterion includes an overlap criterion, wherein the overlap criterion is satisfied based on: generating a graph of a plurality of nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; identifying a reference node associated with a known reference state from the plurality of nodes; and selecting all nodes from the plurality of nodes that share a partition with the reference node.

[0385] Terms and Conditions 92 : The method according to any one of clauses 81 to 94, wherein the including criterion includes an overlap criterion, wherein the overlap crit...

Claims

1. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K Each basis vector state is used to define the core space of the chemical system, wherein: The proposed space consists of a subset of basis states from the complete basis space of the chemical system, which can be sampled from a single parameterized quantum circuit on a quantum computer system. The complete basis space consists of each basis state describing the chemical system. Select according to the selection protocol. K A number of basis vector states are used to define the core space of the chemical system, wherein the selection protocol uses, in addition to fixed... K In addition to the maximum value selection protocol, at least one selection criterion identifies each basis state to be included in the core space from the proposed space, in the fixed... K In the maximum value selection protocol, each basis state is selected as the K highest probability basis state in the proposed space based on its probability of being among the top K highest probabilities. K One of the basis vector states; The characteristic solutions of the chemical system are calculated from the core space via a characteristic solver; and Output the characteristic solution of the chemical system.

2. The method of claim 1, wherein the basis states are selected from the list before any sorting operation is performed on the list of basis states included in the proposed space. K basis vector states, wherein K Each basis state is selected based on a probability value higher than a probability threshold.

3. The method of claim 2, wherein the number of the K basis states that are above the probability threshold to be returned in the list before selection is unknown.

4. The method according to any one of claims 1 to 3, wherein the selected... K The basis states also include: Select an initial set from the proposed space. K selected Each basis vector state defines a first portion of the core space of the chemical system, and is fewer in number than a set defining the core space. K desired Each basis vector state; and Select a complementary set from the symmetry space of the chemical system. K supplemental Several basis states are used to define the remainder of the core space, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system, and each of the supplementary basis states is not part of the core space. K selected Members of the basis vector states, and K supplemental + K selected = K desired .

5. The method according to any one of claims 1 to 4, wherein the selected... K The basis states also include: Select an initial set from the proposed space. K selected basis vector states, wherein K selected Each basis vector defines the core space of the chemical system, and its number is greater than the set of basis vectors that define the core space. K desired One basis vector state; From the set K selected Choose a set of basis states to remove. K removal There are basis states, among which K selected - K removal = K desired ;as well as From the above K selected Remove the aforementioned from each of the basis vector states K removal Each basis vector state is used to generate the set of eigenvalues ​​from which the characteristic solution is computed. K desired Each basis vector state.

6. The method according to any one of claims 1 to 5, wherein the selection is based on an electron conservation selection protocol. K Each basis vector state is used to define the core space; Wherein, when a given basis state from the proposed space is represented by a bit string containing a number of "1"s equal to the number of electrons considered when calculating the characteristic solution of the chemical system, the electron conservation selection protocol selects the given basis state to include in the... K In each of the basis vector states.

7. The method according to any one of claims 1 to 5, wherein the electrons are selected according to the α-β electron conservation selection protocol. K A number of basis vector states are used to define the core space of the chemical system; Wherein, when a given basis state from the proposed space is represented by a bit string containing the number of "1"s at the corresponding α position equal to the number of electrons in the α orbital, and the number of "1"s at the corresponding β position equal to the number of electrons in the β orbital considered when calculating the characteristic solution of the chemical system, the α-β electron conservation selection protocol selects the given basis state to include in the proposed space. K In each of the basis vector states.

8. The method according to any one of claims 1 to 5, wherein the selection is based on an overlapping partition selection protocol with partition bias. K A number of basis states are used to define the core space of the chemical system, wherein the overlap partitioning selection protocol selects a given basis state from the proposed space to be included in the... K In each basis vector state: Construct a graph comprising multiple nodes, wherein each node corresponds to a basis state of the proposed space; Multiple edges are added between the plurality of nodes in the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; and Based on a partitioning bias, all nodes connected to the reference node via one or more edges are selected from the plurality of nodes, wherein the partitioning bias omits any nodes among the plurality of nodes that cannot be linked to the reference node via one or more edges.

9. The method according to any one of claims 1 to 5, wherein the selection is based on an overlapping partition selection protocol with a breadth-first bias. K A number of basis states are used to define the core space, wherein the overlapping partition selection protocol selects a given basis state from the proposed space to be included in the core space in such a way as follows: K In each basis vector state: Construct a graph comprising multiple nodes, wherein each node corresponds to a basis state of the proposed space; Multiple edges are added between the plurality of nodes in the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; and Based on a breadth-first search bias, nodes are selected from the plurality of nodes, starting from the reference node. This breadth-first search bias prioritizes nodes that are fewer edges away from the reference node, rather than nodes that are more edges away, until a node has been selected. K A node or all edges have been traversed.

10. The method according to any one of claims 1 to 5, wherein the selection is based on an overlapping partition selection protocol with optimal priority bias. K A number of basis states are used to define the core space, wherein the overlapping partition selection protocol selects a given basis state from the proposed space to be included in the core space in such a way as follows: K In each basis vector state: Construct a graph comprising multiple nodes, wherein each node corresponds to a basis state of the proposed space; Multiple edges are added between the plurality of nodes in the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; Assign a heuristic score to each node; as well as Based on a best-priority bias, nodes are selected from the plurality of nodes, starting from the reference node and one or more subsequently selected nodes. This best-priority bias selects nodes connected to the reference node or the subsequently selected node by a single edge and having a higher heuristic score value, rather than nodes with a lower heuristic score value, until a node has been selected. K A node or all edges have been traversed.

11. The method according to any one of claims 1 to 5, wherein the selection is made according to a contribution-based iterative selection protocol from among […]. N Selecting from the hypothetical space of the basis vector states K Each basis vector state is used to define the core space, wherein the contribution-based iterative selection protocol is executed in the following manner: Apply the electron conservation selection protocol or the α-β electron conservation selection protocol to generate a set of electron conservation selection protocols or the α-β electron conservation selection protocol that satisfy the electron conservation selection protocol or the α-β electron conservation selection protocol. S There are basis states, among which K < S < N ; From the above S The pair with the highest probability among the basis vector states. K Each basis vector state is experimentally selected to generate a set of... K An experimentally selected basis state and a set of SK One remaining basis state; Using the classical computing system K An experimentally selected basis vector constructs a subspace Hamiltonian. H K ; Hamiltonian from the subspace via the feature solver H K Calculate the characteristic solutions of the chemical system; Based on the feature solution, as described above K The amplitude of each of the experimentally selected basis states From the above K Selected from experimentally chosen basis states M There are basis states, among which M < K The amplitude Indicated by K The degree of contribution of each of the experimentally selected basis states to the characteristic solution; For the SK Each of the remaining basis states is assigned a significance score, the significance score indicating the significance of the remaining basis states. SK The degree of contribution of each of the remaining basis states to the characteristic solution; From the above S Select from the basis states that have the preceding... R The highest significance score R Each basis state, including those not included in the... K At least one of the experimentally selected basis states; as well as The M each basis vector state and the R A set of basis vectors are combined to produce a set of basis vectors. K The evaluated basis states are used to define the core space.

12. The method according to any one of claims 1 to 11, wherein the selected K The basis states also include: Select an initial set from the proposed space. K 1 A number of basis vector states are used to define the initial core space of the chemical system; In response to determining the selected K 1 The number of basis states is insufficient for the feature solver to generate the feature solution of the chemical system from the initial core space, thus determining a solution different from the one obtained from the initial core space. K 1 of K 2 value; as well as Select a set from the proposed space of the chemical system K 2 each of the basis vector states is selected according to the selection protocol. K 2 Each basis vector state is used to define the second core space of the chemical system.

13. The method according to any one of claims 1 to 12, wherein the selected... K The basis states also include: Select an initial set from the proposed space. K One basis state, wherein the initial selection protocol is used to select the basis state. K 1 A number of basis vector states are used to define the initial core space of the chemical system; In response to determining the selected K The number of basis states is insufficient for the characteristic solver to generate characteristic solutions of the chemical system from the initial core space: From selecting the initial set K A set of fundamental vector states of the chemical system is selected from the proposed space. K 2 basis states, wherein the are selected according to a second selection protocol different from the initial selection protocol. K 2 Each basis vector state is used to define the second core space of the chemical system; The initial core space and the second core space are merged to define the core space as the union of the initial core space and the second core space.

14. The method according to any one of claims 1 to 13, wherein the selected... K The basis states also include: According to the selection protocol, an initial set of options is selected from the initial hypothetical space of the chemical system. K One basis state is used to define the initial core space of the chemical system; In response to determining the selected K The number of basis states is insufficient for the feature solver to generate the feature solution of the chemical system from the initial core space: Generate a second hypothetical space corresponding to at least one basis state not included in the initial hypothetical space; Select a set from the second hypothetical space of the chemical system K 2 each of the basis vector states is selected according to the selection protocol. K 2 Each basis vector state is used to define the second core space of the chemical system; as well as The initial core space and the second core space are merged to define the core space as a union core space.

15. The method according to any one of claims 1 to 14, wherein the selected... K The basis states also include: Select from the proposed space according to the selection protocol. K 1 A number of basis vector states are used to define the first part of the core space of the chemical system; as well as Additional basis states outside the proposed space are created by exchanging the α and β values ​​representing the basis states in the proposed space, thereby defining a second part of the core space.

16. The method according to any one of claims 1 to 15, wherein the selection is made according to the selection protocol. K Each basis vector state is used to define the first part of the core space of the chemical system, and the selection also includes: Construct a randomly selected subset of basis states from the symmetry space of the chemical system, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system, and the basis states are not included in the proposed space; Calculate the Hamming distance of the randomly selected subset of basis states; A probability distribution is constructed based on the Hamming distance of the randomly selected subset of basis states; And Sample from the randomly selected subset of basis states according to the probability distribution. M Each basis vector state is used to define the second part of the core space.

17. The method according to any one of claims 1 to 16, wherein: The K Each basis vector is generated from the proposed space of the chemical system. m A set of basis states in a set of basis states, wherein m Each set of basis states in the set includes k There are basis states, among which k For the above m The values ​​of each group of basis states are different; Build m The subspace Hamiltonian, the m Each subspace Hamiltonian in the subspace Hamiltonian corresponds to the... m A set of basis states in a set of basis states; The chemical system is calculated via the feature solver. m Each of the 10 characteristic solutions corresponds to one of the 10 characteristic solutions. m One Hamiltonian in a subspace Hamiltonian; From the classical computer system m Each characteristic solution is used to calculate the weighted sum of the characteristic solutions of the complete Hamiltonian of the chemical system; as well as Output the weighted sum characteristic solution of the chemical system.

18. The method according to any one of claims 1 to 17, wherein the selection according to the selection protocol... K Each basis vector state defines the initial core space of the chemical system, and the method further includes: An extended space is identified by extending the initial core space using a classical computer system based on basis states included in the initial core space, wherein the extended space includes at least one basis state not included in the proposed space; and The initial core space and the extended space are merged to define the core space as the extended core space.

19. The method of claim 18, wherein merging the initial core space and the extended space further comprises: Compressing only one of the initial core space, only the extended space, or only the extended core space, to include no more than K Each basis vector state.

20. The method according to any one of claims 1 to 19, wherein the chemical system selected from the proposed space of the chemical system K The basis states are the first set selected from the first hypothetical space. K The method further includes: a number of basis vector states to define the core space as a first core space; The first characteristic solution of the chemical system is calculated from the first core space via the characteristic solver; In response to determining that the first characteristic solution has not converged: A second state is prepared for the test wave function of the chemical system, wherein the structure of the quantum circuit used to prepare the second state is altered relative to the structure used to prepare the first state. A second hypothetical space is generated from the experimental wavefunction based on the second state via the quantum computer system, wherein the second hypothetical space comprises fewer basis states than the complete basis space of the chemical system; A second group is selected from the second proposed space according to the selection protocol. K A number of basis vector states are used to define the second core space of the chemical system; The second characteristic solution of the chemical system is calculated from the second core space via the characteristic solver; as well as In response to determining that the second characteristic solution has indeed converged, the second characteristic solution of the chemical system is output.

21. The method according to any one of claims 1 to 20, wherein the fixing K The maximum value selection protocol is used as a secondary selection protocol in conjunction with different primary selection protocols.

22. The method according to any one of claims 1 to 21, further comprising: The chemical system in a state relative to the target biomolecule is simulated using the characteristic solutions via a classical computer system. as well as According to the simulation, the chemical system is used to administer a therapeutically effective dose to a biological subject to treat the condition.

23. The method according to any one of claims 1 to 22, wherein the value of K is selected based on the computing power of a classical computer system, said computing power being used to construct a subspace Hamiltonian comprising K basis vector states in a K×K matrix to represent the complete Hamiltonian of the chemical system.

24. A classical computing system comprising a processor and a memory, the classical computing system being configured to perform the method according to any one of claims 1 to 23.

25. A hybrid computing system, the hybrid computing system comprising the classical computing system and the quantum computing system as described in claim 24.

26. The hybrid computing system of claim 25, wherein the feature solver is provided by one or both of the classical computing system and the quantum computing system.

27. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: From the proposed space of the chemical system, a first set of basis states is selected to define the first core space, where: The proposed space consists of a subset of basis states from the complete basis space of the chemical system, which can be sampled from a single parameterized quantum circuit on a quantum computer system. The complete basis space consists of each basis state describing the chemical system. The basis vector states are selected for the first group according to the first selection protocol to define the first core space of the chemical system; The first characteristic solution of the chemical system is calculated from the first core space via a characteristic solver; Determine whether the first characteristic solution converges; In response to determining that the first characteristic solution has not converged: A second set of basis vector states is selected to define the second core space of the chemical system; The second characteristic solution of the chemical system is calculated from the core space via the characteristic solver; and In response to the convergence of the second characteristic solution, the second characteristic solution of the chemical system is output.

28. The method of claim 27, wherein the first number of basis states selected for the first group is different in number from the second number of basis states selected for the second group.

29. The method according to any one of claims 27 to 28, wherein the first number of basis states selected for the first group is equal in number to the second number of basis states selected for the second group.

30. The method according to any one of claims 27 to 29, wherein the first selection protocol is selected from the following: Electronic conservation selection protocol; α-β electron conservation selection protocol; Overlapping partition selection protocols with partition bias; Overlapping partition selection protocol with breadth-first bias; An overlapping partitioning selection protocol with optimal priority bias; Contribution-based iterative selection protocol; fixed K Maximum value selection protocol; and A protocol for selecting from a list of unsorted basis states included in the proposed space, based on probability values ​​that are above a probability threshold.

31. The method according to any one of claims 27 to 30, wherein the second group is selected according to the first selection protocol.

32. The method according to any one of claims 27 to 31, wherein the second group is selected according to a second selection protocol different from the first selection protocol.

33. The method according to any one of claims 27 to 32, wherein selecting the first set of basis states further comprises: Select an initial set from the proposed space. K selected Each basis vector state defines a first portion of the first core space of the chemical system, and is fewer in number than a set defining the first core space. K desired Each basis vector state; and Select a complementary set from the symmetry space of the chemical system. K supplemental Several basis states are used to define the remainder of the first core space, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system, and wherein each of the supplementary basis states is not one of the... K selected Members of the basis vector states, and K supplemental + K selected = K desired .

34. The method according to any one of claims 27 to 33, wherein selecting the first set of basis states further comprises: Select an initial set from the proposed space. K selected basis vector states, wherein K selected Each basis vector defines the first core space of the chemical system, and its number is greater than the set of base vectors that define the core space. K desired One basis vector state; From the set of K selected Choose a set of basis states to remove. K removal There are basis states, among which K selected - K removal + K desired ;as well as From the K selected Remove the aforementioned from each of the basis vector states K removal Each basis vector state is used to generate the set of basis vector states from which the first characteristic solution is computed. K desired Each basis vector state.

35. The method according to any one of claims 27 to 34, wherein selecting the second set of basis states further comprises: A second hypothetical space is generated, which is different from the hypothetical space from which the first set of basis states is selected.

36. The method according to any one of claims 27 to 35, wherein the second set of basis states is selected from the proposed space.

37. The method according to any one of claims 27 to 36, wherein: The first set of basis states and the second set of basis states are generated from the proposed space of the chemical system. m Two sets of basis states in a set of basis states, wherein m Each set of basis states in the set includes k There are basis states, among which k For the above m The values ​​of each group of basis states are different; Build m The subspace Hamiltonian, the m Each subspace Hamiltonian in the subspace Hamiltonian corresponds to the... m A set of basis states in a set of basis states; The chemical system is calculated via the feature solver. m Each of the 10 characteristic solutions corresponds to one of the 10 characteristic solutions. m One of the Hamiltonians in the subspace Hamiltonians, the first characteristic solution and the second characteristic solution are the Hamiltonians of the subspace Hamiltonians. m Two characteristic solutions among the characteristic solutions; From the classical computer system m Each characteristic solution is used to calculate the weighted sum of the characteristic solutions of the complete Hamiltonian of the chemical system; as well as Output the weighted sum characteristic solution of the chemical system.

38. The method according to any one of claims 27 to 37, the method further comprising: An extended space is identified by extending the first core space using a classical computer system based on basis states included in the first core space, wherein the extended space includes at least one basis state not included in the proposed space; and The first core space and the extended space are merged to define the first core space as the extended core space.

39. The method according to any one of claims 27 to 38, further comprising: Select from the proposed space K 1 A number of basis vector states are used to define a first portion of the first core space of the chemical system; Additional basis states outside the proposed space are created by exchanging the α and β values ​​representing the basis states in the proposed space, thereby defining a second part of the core space; as well as When calculating the first feature solution, it includes the first part and the second part of the first core space.

40. The method according to any one of claims 27 to 39, further comprising: Select from the proposed space K A number of basis vector states are used to define the first part of the core space of the chemical system; Construct a randomly selected subset of basis states from the symmetry space of the chemical system, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system, and the basis states are not included in the proposed space; Calculate the Hamming distance of the randomly selected subset of basis states; A probability distribution is constructed based on the Hamming distance of the randomly selected subset of basis states; Sample from the randomly selected subset of basis states according to the probability distribution. M A number of basis vector states are used to define the second part of the core space; as well as When calculating the first feature solution, it includes the first part and the second part of the first core space.

41. The method according to any one of claims 27 to 40, further comprising: The chemical system in a state relative to the target biomolecule is simulated using the second characteristic solution via a classical computer system. as well as According to the simulation, the chemical system is used to administer a therapeutically effective dose to a biological subject to treat the condition.

42. The method according to any one of claims 27 to 41, wherein the number of basis states in the first group is K1, wherein the value of K1 is selected based on the computing power of a classical computer system, the computing power being used to construct a subspace Hamiltonian comprising K1 basis states in a K1×K1 matrix to represent the complete Hamiltonian of the chemical system.

43. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select the first group from the hypothetical space of chemical systems. K One basis state, The hypothetical space consists of a subset of basis states from the complete basis space of the chemical system, which can be sampled from a single parameterized quantum circuit on a quantum computer system. The complete basis space is composed of each basis state describing the chemical system. The selection is based on the selection protocol. K A number of basis vector states are used to define the core space of the chemical system; The characteristic solutions of the chemical system are calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

44. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select an initial set from the hypothetical space of the chemical system. K selected A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K selected A number of basis vector states are used to define the first part of the core space of the chemical system, and are fewer in number than a set of basis vector states used to define the core space to compute the characteristic solutions of the chemical system via the characteristic solver. K desired One basis vector state; Select a complementary set from the symmetry space of the chemical system. K supplemental A number of basis states are used to define the remainder of the core space, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system. K supplemental Each of the basis states is not the one described. K selected Members of the basis vector states, and K supplemental + K selected = K desired ; The characteristic solutions of the chemical system are calculated from the core space via the characteristic solver; as well as Output the characteristic solution of the chemical system.

45. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select an initial set from the hypothetical space of the chemical system. K selected A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K selected A number of basis vector states are used to define the first part of the core space of the chemical system, and are greater in number than a set of basis vector states used to define the core space to compute the characteristic solutions of the chemical system via the characteristic solver. K desired One basis vector state; From the set K selected Choose a set of basis states to remove. K removal There are basis states, among which K selected - K removal = K desired ; From the above K selected Remove the aforementioned from each of the basis vector states K removal Each basis vector state is used to generate the set of eigenvalues ​​from which the characteristic solution is computed. K desired One basis vector state; The characteristic solutions of the chemical system are calculated from the core space via the characteristic solver; as well as Output the characteristic solution of the chemical system.

46. ​​A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on the electronic conservation selection protocol. K A number of basis vector states are used to define the core space of the chemical system; wherein when the origin is from the N When a given basis state is represented by a bit string containing a number of "1"s equal to the number of electrons considered in calculating the characteristic solution of the chemical system, the electron conservation selection protocol selects the given basis state to include the given basis state in the given basis state. K In each of the basis vector states; The characteristic solution of the chemical system is calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

47. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on the α-β electron conservation selection protocol. K Each basis vector state is used to define the core space of the chemical system, wherein when the vector state originates from the... N When a given basis state is represented by a bit string containing a number of "1"s at the corresponding α position equal to the number of electrons in the α orbital, and a number of "1"s at the corresponding β position equal to the number of electrons in the β orbital considered when calculating the characteristic solution of the chemical system, the α-β electron conservation selection protocol selects the given basis state to include the given basis state in the... K In each of the basis vector states; The characteristic solution of the chemical system is calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

48. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on an overlapping partition selection protocol with partition bias. K A number of basis vector states are used to define the core space of the chemical system, wherein the overlapping partition selection protocol is derived from the following... N A given basis state is selected from the basis states to include the basis states in the... K In each basis vector state: Construct a graph comprising multiple nodes, where each node corresponds to the... N One of the basis states; Multiple edges are added between the plurality of nodes in the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; as well as Based on partition bias, select all nodes from the plurality of nodes that are connected to the reference node via one or more edges, wherein the partition bias omits any nodes from the plurality of nodes that cannot be linked to the reference node via one or more edges in the selection; The characteristic solution of the chemical system is calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

49. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on an overlapping partitioning selection protocol with a breadth-first bias. K A number of basis vector states are used to define the core space of the chemical system, wherein the overlapping partition selection protocol is derived from the following... N A given basis state is selected from the basis states to include the basis states in the... K In each basis vector state: Construct a graph comprising multiple nodes, where each node corresponds to the... N One of the basis states; Multiple edges are added between the plurality of nodes in the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; as well as Based on a breadth-first search bias, nodes are selected from the plurality of nodes, starting from the reference node. This breadth-first search bias prioritizes nodes that are fewer edges away from the reference node, rather than nodes that are more edges away, until a node has been selected. K A node or all edges have been traversed; The characteristic solution of the chemical system is calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

50. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on an overlapping partition selection protocol with the best priority bias. K A number of basis vector states are used to define the core space of the chemical system, wherein the overlapping partition selection protocol is derived from the following... N A given basis state is selected from the basis states to include the basis states in the... K In each basis vector state: Construct a graph comprising multiple nodes, where each node corresponds to the... N One of the basis states; Multiple edges are added between the plurality of nodes in the graph, wherein each of the multiple edges is defined between two nodes in the graph, the two nodes representing two basis states |b1〉 and |b2〉, where 〈b1|H|b2〉 is nonzero; Assign a heuristic score to each node; as well as Based on a best-priority bias, nodes are selected from the plurality of nodes, starting from the reference node and one or more subsequently selected nodes. This best-priority bias selects nodes connected to the reference node or the subsequently selected node by a single edge and having a higher heuristic score value, rather than nodes with a lower heuristic score value, until a node has been selected. K A node or all edges have been traversed; The characteristic solution of the chemical system is calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

51. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N The selection is based on a contribution-based iterative selection protocol. K Each basis vector state is used to define the core space of the chemical system, wherein the contribution-based iterative selection protocol is executed in the following manner: Apply the electron conservation selection protocol or the α-β electron conservation selection protocol to generate a set of electron conservation selection protocols or the α-β electron conservation selection protocol that satisfy the electron conservation selection protocol or the α-β electron conservation selection protocol. S There are basis states, among which K < S < N ; From the above S The pair with the highest probability among the basis vector states. K Each basis vector state is experimentally selected to generate a set of... K An experimentally selected basis state and a set of SK One remaining basis state; Using the classical computing system K An experimentally selected basis vector constructs a subspace Hamiltonian. H K ; Hamiltonian from the subspace via the feature solver H K Calculate the characteristic solutions of the chemical system; Based on the feature solution, as described above K The amplitude of each of the experimentally selected basis states From the above K Selected from experimentally chosen basis states M There are basis states, among which M < K The amplitude Indicated by K The degree of contribution of each of the experimentally selected basis states to the characteristic solution; For the SK Each of the remaining basis states is assigned a significance score, the significance score indicating the significance of the remaining basis states. SK The degree of contribution of each of the remaining basis states to the characteristic solution; From the above S Select from the basis states that have the preceding... R The highest significance score R Each basis state, including those not included in the... K At least one of the experimentally selected basis states; The M each basis vector state and the R A set of basis vectors are combined to produce a set of basis vectors. K The evaluated basis states are used to define the core space; The characteristic solution of the chemical system is calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

52. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K 1,000 basis states, wherein the hypothetical space is generated by a quantum computer system and includes samples from the complete basis space of the chemical system. N There are basis states, among which K < N , among which according to fixed K A maximum value selection protocol is used to select the... K A number of basis vector states are used to define the core space of the chemical system, wherein the fixed vector states are... K The maximum value selection protocol is based on the N The probability of each basis state is related to the N Sort the basis vector states and from the... N The core space is defined by selecting the K basis states with the highest probability from the given basis states. The characteristic solution of the chemical system is calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

53. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select an initial set from the hypothetical space of the chemical system. K 1 A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K 1 A number of basis vector states are used to define the initial core space of the chemical system; In response to determining the selected K 1 The number of basis states is insufficient for the feature solver to generate a feature solution for the chemical system from the core space, thus determining a solution different from the one obtained from the core space. K 1 of K 2 value; Select a set from the proposed space of the chemical system K 2 each of the basis vector states is selected according to the selection protocol. K 2 Each basis vector state is used to define the second core space of the chemical system; The characteristic solutions of the chemical system are calculated from the second core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

54. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select an initial set from the hypothetical space of the chemical system. K One basis state, wherein the proposed space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis state is selected according to an initial selection protocol. K 1 A number of basis vector states are used to define the initial core space of the chemical system; In response to determining the selected K 1 The number of basis vector states is insufficient for the feature solver to generate a feature solution for the chemical system from the initial core space: From selecting the initial set K A set of fundamental vector states of the chemical system is selected from the proposed space. K 2 basis states, wherein the are selected according to a second selection protocol different from the initial selection protocol. K 2 Each basis vector state is used to define the second core space of the chemical system; The initial core space and the second core space are merged to define a union core space; The characteristic solutions of the chemical system are calculated from the union core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

55. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select an initial set from the initial hypothetical space of the chemical system. K One basis state, wherein the initial hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis state is selected according to a selection protocol. K One basis state is used to define the initial core space of the chemical system; In response to determining the selected K The number of basis states is insufficient for the characteristic solver to generate characteristic solutions of the chemical system from the initial core space: Generate a second hypothetical space corresponding to at least one basis state not included in the initial hypothetical space; Select a set from the second hypothetical space of the chemical system K 2 each of the basis vector states is selected according to the selection protocol. K 2 Each basis vector state is used to define the second core space of the chemical system; The initial core space and the second core space are merged to define a union core space; The characteristic solutions of the chemical system are calculated from the union core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

56. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K 1 A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K 1 A number of basis vector states are used to define the first part of the core space of the chemical system; Additional basis states outside the proposed space are created by exchanging the α and β values ​​representing the basis states in the proposed space, thereby defining a second part of the core space; The characteristic solution of the chemical system is calculated from the core space, which includes the first part and the second part, via a characteristic solver; as well as Output the characteristic solution of the chemical system.

57. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K A number of basis vector states are used to define the first part of the core space of the chemical system; Construct a randomly selected subset of basis states from the symmetry space of the chemical system, wherein the symmetry space comprises fewer basis states than the complete basis space of the chemical system, and the basis states are not included in the proposed space; Calculate the Hamming distance of the randomly selected subset of basis states; A probability distribution is constructed based on the Hamming distance of the randomly selected subset of basis states; According to the probability distribution, M basis states are sampled from the randomly selected subset of basis states to define a second part of the core space; The characteristic solution of the chemical system is calculated from the core space, which includes the first part and the second part, via a characteristic solver; as well as Output the characteristic solution of the chemical system.

58. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K A set of basis states is used to define a core space, wherein the proposed space is generated by a quantum computer system and includes fewer basis states than the complete basis space of the chemical system, wherein the proposed space is selected from an unsorted list of basis states included in the proposed space based on probability values ​​having a probability threshold. K One basis vector state; The characteristic solutions of the chemical system are calculated from the core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

59. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Generate from the hypothetical space of the chemical system m The basis vector states, wherein m Each set of basis states in the set includes k There are basis states, among which k The value of each of the m sets of basis states is different; Build m The subspace Hamiltonian, the m Each subspace Hamiltonian in the subspace Hamiltonian corresponds to the... m A set of basis states in a set of basis states; The chemical system is calculated via a feature solver. m Each of the 10 characteristic solutions corresponds to one of the 10 characteristic solutions. m One Hamiltonian in a subspace Hamiltonian; From the classical computer system m Each characteristic solution is used to calculate the weighted sum of the characteristic solutions of the complete Hamiltonian of the chemical system; as well as Output the characteristic solution of the chemical system.

60. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select from the hypothetical space of chemical systems K A number of basis states, wherein the hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the basis states are selected according to a selection protocol. K A number of basis vector states are used to define the initial core space of the chemical system; The initial core space is expanded by a classical computer system based on the basis states included in the initial core space to identify the expanded space, wherein the expanded space includes at least one basis state not included in the proposed space; The initial core space and the extended space are merged to define the extended core space; The characteristic solutions of the chemical system are calculated from the expanded core space via a characteristic solver; as well as Output the characteristic solution of the chemical system.

61. The method of claim 60, wherein merging the initial core space and the extended space further comprises: Compressing only one of the initial core space, only the extended space, or only the extended core space, to include no more than K Each basis vector state.

62. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Select the first group from the first hypothetical space of the chemical system. K A number of basis states, wherein the first hypothetical space is generated by a quantum computer system and comprises fewer basis states than the complete basis space of the chemical system, wherein the first state is selected from the first hypothetical space according to a selection protocol. K A number of basis vector states are used to define the core space of the chemical system; The first characteristic solution of the chemical system is calculated from the first core space via a characteristic solver; In response to determining that the first characteristic solution has not converged: A second state is prepared for the test wave function of the chemical system, wherein the structure of the quantum circuit used to prepare the second state is altered relative to the structure used to prepare the first state. A second hypothetical space is generated from the experimental wavefunction based on the second state via the quantum computer system, wherein the second hypothetical space comprises fewer basis states than the complete basis space of the chemical system; A second group is selected from the second proposed space according to the selection protocol. K A number of basis vector states are used to define the second core space of the chemical system; The second characteristic solution of the chemical system is calculated from the second core space via the characteristic solver; as well as In response to determining that the second characteristic solution has indeed converged, the second characteristic solution of the chemical system is output.

63. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Created via a classical computer system with 2 n The complete Hamiltonian of the Hilbert space of the basis vector states; A first series of iterations is run on a hybrid quantum-classical computer system comprising the classical computer system and the quantum computer system until a first stopping criterion is met, wherein each iteration in the first series of iterations includes: An initial state is assigned to the complete Hamiltonian via the classical computer system and via an initial state preparation protocol; A test wavefunction is generated from the initial state using the test wavefunction preparation protocol via the hybrid quantum-classical computer system. A second series of iterations is run on the hybrid quantum-classical computer system until a second stopping criterion is met, wherein each iteration in the second series of iterations includes: The batch results, including values ​​from the experimental wavefunction, are sampled via the quantum computer system, wherein: The value represents the computed basis state of the Hilbert space; The second multiple iterations are stopped on the hybrid quantum-classical computer system in response to the satisfaction of the second stopping criterion, wherein a first plurality of values ​​are output, the first plurality of values ​​including the batch results sampled in each iteration of the second multiple iterations; via the classical computer system, a second plurality of values ​​satisfying a criterion are selected from the first plurality of values, the selection being performed by selecting values ​​from the first plurality of values ​​according to a selection protocol, wherein: The number of values ​​of the second plurality of values ​​is equal to K ; K Less than 2 n ;and The second plurality of values K The values ​​describe the 2 n The basis state with the largest amplitude value K A subset of basis vector states; The selection protocol includes one of the following: Symmetry standard; Overlapping criteria; and Contribution-based iterative criteria; The classical computer system uses the second plurality of values ​​to construct the subspace Hamiltonian representation, including... K The subspace Hamiltonian of each basis vector state is used to represent the complete Hamiltonian; The minimum eigenvalue of the Hamiltonian of the subspace is calculated via a feature solver provided by the hybrid quantum-classical computer system; The eigenvector corresponding to the minimum eigenvalue is computed via the eigenvalue solver provided by the hybrid quantum-classical computer system; and Using the classical computer system, the approximate ground state value representing the ground state of the complete Hamiltonian, and the approximate ground state energy value representing the ground state energy of the complete Hamiltonian, are calculated using the minimum eigenvalue of the subspace Hamiltonian and the eigenvector corresponding to the minimum eigenvalue of the subspace Hamiltonian; and In response to the satisfaction of the first stopping criterion, the first multiple iterations are stopped on the classical computer system; and The output is the ground state value and the ground state energy value from the final iteration of the first plurality of iterations.

64. The method of claim 63, wherein the initial state preparation protocol comprises: The initial state is assigned to the complete Hamiltonian via the classical computer system, wherein the initial state includes at least one of the following: The state prepared via the Hartree-Fock protocol; Zero state; The computational basis states of the Hilbert space; The state prepared via the ab initio initial state preparation protocol; The state prepared via the tensor network initial state preparation protocol; The sparse initial state resulting from the ground state value of the final iteration from the previous first iteration; Uniform distribution state; and Random distribution state.

65. The method according to any one of claims 63 to 64, wherein the experimental wavefunction preparation protocol comprises: The third and subsequent iterations are run via the variational quantum characteristic solver (VQE) on the hybrid quantum-classical computer system until the third stopping criterion is met, wherein each iteration in the third and subsequent iterations includes: A parameterized quantum circuit is constructed using the classical computer system based on the initial state. A parameterized wavefunction approximating the ground state of the complete Hamiltonian is generated by executing the parameterized quantum circuit on the quantum computer system. Pauli sampling is performed on the parameterized wavefunction generated by the parameterized quantum circuit via the quantum computer system with respect to the complete Hamiltonian; The expected value of the parameterized wavefunction relative to the complete Hamiltonian is calculated via the classical computer system; and Update a set of parameters of the parameterized quantum circuit based on the expected value; The third multiple iteration is stopped on the hybrid quantum-classical computer system in response to the satisfaction of the third stopping criterion, wherein the parameterized wavefunction from the last iteration of the third multiple iteration is output; and The third iteration is performed before the second iteration.

66. The method according to any one of claims 63 to 65, wherein the third stopping criterion is satisfied in response to at least one of the following: The predefined number of iterations in the third and subsequent iterations were executed; The third iteration is run for a predefined amount of time. The quantum computer system was used to achieve a predefined amount of time; The change in the expected value from a given iteration in the third or more iterations to subsequent iterations in the third or more iterations is within a threshold value of the quantum noise change in the quantum computer system; The first derivative of the expected value from a given iteration in the third or subsequent iterations falls below the termination threshold; The change in the expected value from multiple previous iterations in the third multiple iterations to subsequent iterations in the third multiple iterations is within the termination threshold; as well as A predefined number of samples across the third iteration, wherein the samples include: The wave function is obtained by executing quantum circuitry via the quantum computer system. The measurement result is obtained by performing a measurement on the wave function via the quantum computer system; and Output the measurement results.

67. The method according to any one of claims 63 to 66, wherein the experimental wavefunction preparation protocol comprises: A parameterized quantum circuit is constructed using the classical computer system based on the initial state. A parameterized wavefunction approximating the ground state of the complete Hamiltonian is generated by executing the parameterized quantum circuitry on the quantum computer system, without performing Pauli sampling; Output the parameterized wavefunction.

68. The method according to any one of claims 63 to 67, wherein the second stopping criterion is satisfied in response to at least one of the following: The second set of iterations was performed a predefined number of times; and The batch results contain a predefined number of values.

69. The method according to any one of claims 63 to 68, wherein the inclusion criterion is satisfied in response to at least one of the following: The second plurality of values ​​comprises a predefined number of values ​​derived from the first plurality of values; and The number of values ​​included in the second plurality of values ​​is within a predefined threshold of the number of values ​​derived from the first plurality of values.

70. The method according to any one of claims 63 to 69, wherein the selection protocol comprises: When the selection criteria result in not exactly K values ​​being selected from the first plurality of values, an additional selection criterion is chosen to select the exact K values ​​for constructing the subspace Hamiltonian, wherein the additional selection criterion is chosen from at least one of the following: Second symmetry standard; Second screening criterion; Overlapping standards; Contribution-based iterative criteria; Randomized selection; Screening criteria; or Choose from the third set of values ​​derived from the previous second or more iterations.

71. The method according to any one of claims 63 to 70, wherein the symmetry criterion is satisfied in response to the value representing a calculated basis state of the Hilbert space having a predetermined number of electrons, or wherein the screening criterion is satisfied in response to the value representing a calculated basis state of the Hilbert space and the amplitude value being within a predefined highest percentage of a plurality of amplitude values ​​of each of the first plurality of values.

72. The method according to any one of claims 63 to 71, wherein the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K The value is represented by a bit string having a number of "1"s equal to the number of electrons in the chemical system represented by the complete Hamiltonian.

73. The method according to any one of claims 63 to 72, wherein the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K The value is represented by a bit string whose number of "1"s at positions corresponding to a given orbit is equal to the number of electrons in the given orbit of the chemical system represented by the complete Hamiltonian.

74. The method according to any one of claims 63 to 73, wherein the overlap criterion is satisfied based on the following: Generate a graph of multiple nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; Identify the reference node associated with the known reference state from among the plurality of nodes; as well as Select all nodes that share a partition with the reference node from among the plurality of nodes.

75. The method according to any one of claims 63 to 74, wherein the overlap criterion is satisfied based on the following: Generate a graph of multiple nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; Identify the reference node associated with the known reference state from the plurality of nodes; and From the plurality of nodes, select all nodes that are a distance from the reference node or a previously selected node by an edge, until all nodes from the plurality of nodes that share a partition with the reference node have been selected, or until at least [number missing] nodes from the plurality of nodes have been selected. K Each node is determined by the first occurrence.

76. The method according to any one of claims 63 to 75, wherein the overlap criterion is satisfied based on the following: Generate a graph of multiple nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; A score is assigned to each of the plurality of nodes based on a heuristic measure of the importance of the corresponding basis states; Identify the reference node associated with the known reference state from among the plurality of nodes; as well as From the plurality of nodes, select the next node that is an edge distance from the reference node or a previously selected node, the next node having the highest score relative to all other nodes in the plurality of nodes that are an edge distance from the reference node or any previously selected node in the plurality of nodes, until all nodes in the plurality of nodes that share a partition with the reference node have been selected, or until at least [number missing] nodes have been selected from the plurality of nodes. K Each node is determined by the first occurrence.

77. The method according to any one of claims 63 to 76, wherein the screening criteria are satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K The value exceeds a predefined threshold.

78. The method according to any one of claims 63 to 77, wherein the contribution-based iteration criterion is satisfied based on the following: A fourth set of values ​​is obtained by applying a symmetry standard to the first plurality of values ​​via the classical computing system. The fourth plurality of values ​​are sorted based on the first plurality of magnitude squared values ​​by the classical calculation system. The fifth plurality of values ​​are selected from the fourth plurality of values ​​via the classical computing system, wherein: The number of values ​​for the fifth plurality of values ​​is equal to k; The subspace Hamiltonian is constructed using the fifth plurality of values ​​via the classical computing system. The characteristic solutions of the subspace Hamiltonian constructed using the fifth plurality of values ​​are obtained via the classical computing system. The classical computing system selects a sixth plurality of values ​​based on a second inclusion criterion, wherein the second inclusion criterion is satisfied based on the following: The seventh value with the largest amplitude is selected from the fifth plurality of values ​​via the classical calculation system; The classical calculation system assigns a first plurality of significance scores to the values ​​in the fourth plurality of values ​​that are not included in the fifth plurality of values, wherein the significance scores in the first plurality of significance scores include: The effect of adding corresponding values ​​to the characteristic solutions of the fifth plurality of values ​​via the classical computer system; The eighth plurality of values ​​are selected via the classical computing system, and the eighth plurality of values ​​include: The number of values ​​for the eighth plurality of values ​​is equal to R; and The value among the eighth plurality of values ​​has a significance score within the first R significance scores; The ninth or more values ​​are combined via the classical computing system, and the ninth or more values ​​include: The sixth multiple values; and The eighth set of values; The ninth or more values ​​are output via the classical computing system.

79. The method according to any one of claims 63 to 78, wherein the first stopping criterion is satisfied in response to at least one of the following: The first set of iterations was performed a predefined number of times. The first multiple iterations are completed within a predefined timeframe. The quantum computer system was used to achieve a predefined amount of time; The change in the ground state energy value from a given iteration in the first multiple iterations to subsequent iterations in the first multiple iterations is within a threshold value of the quantum noise change value of the quantum computer system; The first derivative of the ground state energy value from a given iteration in the first plurality of iterations decreases below the termination threshold; and The change in the ground state energy value from multiple previous iterations in the first multiple iterations to subsequent iterations in the first multiple iterations is within the termination threshold.

80. The method of any one of claims 63 to 79, further comprising, before outputting from the final iteration of the second plurality of iterations the first plurality of values ​​comprising the batch results sampled in each of the second plurality of iterations: In response to the determination that the second stopping criterion is not met: Via the quantum computer system, a second batch of results, including a second value from the experimental wavefunction, is sampled in the computational basis, wherein: The second value represents the computed basis state of the Hilbert space.

81. The method according to any one of claims 63 to 80, further comprising, before outputting the parameterized wavefunction from the last iteration of the third plurality of iterations: In response to the determination that the third stopping criterion is not met: The second parameterized quantum circuit is constructed via the classical computer system based on the initial state; A second parameterized wavefunction approximating the ground state of the complete Hamiltonian is generated by executing the second parameterized quantum circuit on the quantum computer system. Pauli sampling is performed on the second parameterized wavefunction generated by the second parameterized quantum circuit via the quantum computer system with respect to the complete Hamiltonian; The second expectation value of the second parameterized wavefunction relative to the complete Hamiltonian is calculated via the classical computer system. as well as The second set of parameters of the second parameterized quantum circuit is updated based on the second expected value.

82. The method according to any one of claims 63 to 81, further comprising, before outputting the ground state value and the ground state energy value from the final iteration of the first plurality of iterations: In response to determining that the first stopping criterion is not met: A second initial state is assigned to the complete Hamiltonian via the classical computer system and via the initial state preparation protocol; and The first multiple iterations are rerun on the hybrid quantum-classical computer system until the first stopping criterion is met for the second time.

83. The method according to any one of claims 63 to 82, further comprising, before outputting the ground state value and the ground state energy value from the final iteration of the first plurality of iterations: In response to determining that the first stopping criterion is not met: The second iteration is rerun on the quantum computer system until the second stopping criterion is met for the second time.

84. The method according to any one of claims 63 to 83, the method further comprising, before outputting the ground state value and the ground state energy value from the final iteration of the first plurality of iterations: In response to determining that the first stopping criterion is not met: via the classical computer system, new second plurality of values ​​that satisfy the criteria are reselected from the first plurality of values, the reselection being performed by selecting values ​​from the first plurality of values ​​according to the selection protocol.

85. A method for improving computational system efficiency and accuracy in calculating Hamiltonian characteristic solutions, the method comprising: Created via a classical computer system with 2 n The complete Hamiltonian of the Hilbert space of the basis vector states; The initial state is assigned to the complete Hamiltonian via the classical computer system. A parameterized quantum circuit is constructed using the classical computer system based on the initial state. On a quantum computer system, the parameterized quantum circuit is used to run multiple iterations of a variational quantum characteristic solver (VQE) to generate a parameterized wavefunction approximating the ground state of the complete Hamiltonian, wherein each of the multiple iterations includes: Regarding the complete Hamiltonian, Pauli sampling or Pauli measurement is performed on the quantum state generated by the parameterized quantum circuit to calculate the expectation value of the parameterized wavefunction relative to the complete Hamiltonian; and Update the parameters of the parameterized wavefunction based on the expected value; The multiple iterations are stopped on the quantum computer system in response to the satisfaction of the stopping criteria, wherein the parameterized wavefunction from the last iteration of the multiple iterations is output as the finalized wavefunction; Multiple values ​​satisfying a set of criteria are selected from the finalized wavefunction via the classical computer system, wherein: The number of samples in the plurality of samples is equal to K ; K Less than 2 n ;and The multiple values K The values ​​describe the values ​​from the 2 n The basis state with the largest value K A subset of the basis vector states serves as an approximate representation of the energy states in the complete Hamiltonian; Response to determination K Within the computational capabilities of the classical computer system, the computational capabilities are used to... K × K Constructing a matrix including K The subspace Hamiltonians of the basis vector states are used to represent the complete Hamiltonian: Calculated via the classical computer system K Classical values ​​of basis vector states; Using the classic computer system K The classical values ​​of the basis vector states are used to construct the Hamiltonian of the subspace; The minimum eigenvalue of the subspace Hamiltonian is calculated via a feature solver provided by the classical computer system. The classical computer system calculates the eigenvector corresponding to the minimum eigenvalue; and Using the classical computer system, the ground state value representing the ground state and the ground state energy value representing the ground state energy of the complete Hamiltonian are calculated using the minimum eigenvalue and the eigenvector of the subspace Hamiltonian.

86. The method according to claim 85, further comprising: Using the classical computer system, the ground state value and the ground state energy value are used to simulate a chemical system in a state relative to a target biomolecule, which can be represented as 2 n ×2 n The matrix corresponds to the complete Hamiltonian; and According to the simulation, the chemical system is used to administer a therapeutically effective dose to a biological subject to treat the condition.

87. The method according to any one of claims 85 to 86, wherein the stopping criterion is satisfied in response to one of the following: The change in the expected value from a given iteration in the multiple iterations to subsequent iterations in the multiple iterations is within a threshold value of the quantum noise change in the quantum computer system; or The first derivative of the expected value across the multiple iterations decreases below the termination threshold.

88. The method according to any one of claims 85 to 87, the method further comprising determining K Before the computing power of the classical computer system is within its range: The earlier ones determined after the stopping criteria have been met for earlier iterations. K The value is less than the computing power of the classical computer system. A second initial state is assigned to the complete Hamiltonian via the classical computer system, based on the classical ground state and the classical ground state energy. The second parameterized quantum circuit is constructed via the classical computer system based on the second initial state; The VQE is run a second series of iterations on the quantum computer system using the second parameterized quantum circuitry to generate a second parameterized wavefunction of the ground state that approximates the complete Hamiltonian, until the stopping criterion is met; and The classical computer system selects multiple values ​​from the finalized wavefunction that satisfy the criteria, and then reselects the multiple values ​​from the second parameterized wavefunction, such that... K Larger than the earlier one K value.

89. The method according to any one of claims 85 to 88, the method further comprising determining K Before the computing power of the classical computer system is within its range: The earlier ones determined after the stopping criteria have been met for earlier iterations. K The value is greater than the computing power of the classical computer system. A second initial state is assigned to the complete Hamiltonian via the classical computer system, based on the classical ground state and the classical ground state energy. The second parameterized quantum circuit is constructed via the classical computer system based on the second initial state; The VQE is run a second series of iterations on the quantum computer system using the second parameterized quantum circuitry to generate a second parameterized wavefunction of the ground state that approximates the complete Hamiltonian, until the stopping criterion is met; and The classical computer system selects multiple values ​​from the finalized wavefunction that satisfy the criteria, and then reselects the multiple values ​​from the second parameterized wavefunction, such that... K Smaller than the earlier one K value.

90. The method according to any one of claims 85 to 89, wherein the inclusion standard specifies that the plurality of values ​​are identified by the quantum computing system as having non-zero values. n It consists of 1 basis vector state.

91. The method according to any one of claims 85 to 90, wherein the inclusion standard is based on a selection protocol, the selection protocol comprising: When the selection criteria result in not exactly K values ​​being selected from the first plurality of values, an additional selection criterion is chosen to select the exact K values ​​for constructing the subspace Hamiltonian, wherein the additional selection criterion is chosen from at least one of the following: Second symmetry standard; Second screening criterion; Overlapping standards; Randomized selection; Screening criteria; or Choose from the third set of values ​​derived from the previous second or more iterations.

92. The method according to any one of claims 85 to 91, wherein the including criteria include a symmetry criterion, wherein the symmetry criterion is satisfied in response to the value representing a calculated basis state of the Hilbert space having a predetermined number of electrons, or wherein the screening criterion is satisfied in response to the value representing a calculated basis state of the Hilbert space and the amplitude value being within a predefined highest percentage of a plurality of amplitude values ​​of the value among the first plurality of values.

93. The method according to any one of claims 85 to 92, wherein the including criterion includes a symmetry criterion, wherein the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K The value is represented by a bit string having a number of "1"s equal to the number of electrons in the chemical system represented by the complete Hamiltonian.

94. The method according to any one of claims 85 to 93, wherein the including criterion includes a symmetry criterion, wherein the symmetry criterion is satisfied based on the following: the number of values ​​selected for the second plurality of values ​​is equal to K The value is represented by a bit string whose number of "1"s at positions corresponding to a given orbit is equal to the number of electrons in the given orbit of the chemical system represented by the complete Hamiltonian.

95. The method according to any one of claims 85 to 94, wherein the including criterion includes an overlap criterion, wherein the overlap criterion is satisfied based on the following: Generate a graph of multiple nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; Identify the reference node associated with the known reference state from among the plurality of nodes; as well as Select all nodes that share a partition with the reference node from among the plurality of nodes.

96. The method according to any one of claims 85 to 95, wherein the including criterion includes an overlap criterion, wherein the overlap criterion is satisfied based on the following: Generate a graph of multiple nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; Identify the reference node associated with the known reference state from the plurality of nodes; and From the plurality of nodes, select all nodes that are a distance from the reference node or a previously selected node by an edge, until all nodes from the plurality of nodes that share a partition with the reference node have been selected, or until at least [number missing] nodes from the plurality of nodes have been selected. K Each node is determined by the first occurrence.

97. The method according to any one of claims 85 to 96, wherein the including criterion includes an overlap criterion, wherein the overlap criterion is satisfied based on the following: Generate a graph of multiple nodes from the first plurality of values, wherein each pair of nodes such that <b1|H|b2> is nonzero is connected by an edge; A score is assigned to each of the plurality of nodes based on a heuristic measure of the importance of the corresponding basis states; Identify the reference node associated with the known reference state from among the plurality of nodes; as well as From the plurality of nodes, select the next node that is an edge distance from the reference node or a previously selected node, the next node having the highest score relative to all other nodes in the plurality of nodes that are an edge distance from the reference node or any previously selected node in the plurality of nodes, until all nodes in the plurality of nodes that share a partition with the reference node have been selected, or until at least [number missing] nodes have been selected from the plurality of nodes. K Each node is determined by the first occurrence.

98. The method according to any one of claims 85 to 97, wherein the including criteria include a screening criterion, wherein the screening criterion is satisfied based on the following: the number of values ​​selecting the second plurality of values ​​is equal to K The value exceeds a predefined threshold.

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