Quantum circuit evolution via krylov subspaces for hybrid computing of hamiltonian eigensolutions

Abstract

Calculation control for hybrid computing of Hamiltonian eigensolutions may be provided by preparing a plurality of states in a Krylov subspace of a full Hamiltonian that represents a chemical system; sampling a plurality of basis states from each state of the plurality of states in the Krylov subspace to produce a plurality of sampled basis states; processing the plurality of sampled basis states to yield an analysis selection; constructing a subspace Hamiltonian from the analysis selection; computing, via an eigensolver, an eigensolution for the chemical system from the subspace Hamiltonian; and outputting the eigensolution for the chemical system.

Description

QUANTUM CIRCUIT EVOLUTION VIA KRYLOV SUBSPACES FOR HYBRID COMPUTING OF HAMILTONIAN EIGENSOLUTIONSCROSS-REFERENCES TO RELATED APPLICATIONS

[0001] The present disclosure claims benefit of U.S. Provisional Application Serial No.: 63 / 709,890 titled “QUANTUM CIRCUIT EVOLUTION VIA KRYLOV SUBSPACES FOR HYBRID COMPUTING OF HAMILTONIAN EIGENSOLUTIONS”, which was filed on 2024-10-21, and is incorporated by reference herein in its entirety.BACKGROUNDField

[0002] The present disclosure relates to computation using quantum computers.Quantum Computers

[0003] Quantum computers, in contrast to classical computers, use quantum mechanical phenomena to store data in quantum bits or “qubits” that can store classical representation of ‘O’, ‘1’ or the superposition of the two. Accordingly, rather than outputting true / false or 1 / 0 when read as a classical bit would provide, a qubit provides these classical values as output probabilistically based on the state of the qubit. Quantum computers have several promising uses that benefit from the nondeterministic nature of quantum phenomena, which are capable of providing greater speed and efficiency in solving certain classes of calculations than classical computers can provide. However, these quantum computers suffer from noise in those calculations that can prevent the quantum computers from reaching solutions for various calculations (e.g., being trapped in local minima / maxima, having noise dominate an output value, etc.).Calculating Hamiltonians

[0004] One field where quantum computers are showing promise is in the simulation of the quantum properties of chemical systems, such as by calculating the eigensolutions of the Hamiltonian of a system. The Hamiltonian corresponds to the total energy (e.g., the kinetic and potential energies of the constituent particles) in the simulated system and provides the possible energies according to a set of eigenvalues and eigenvectors. These values have proven ofinterest for researchers in evaluating potential chemical interactions of complex molecules with one another and biological structures (e.g., cells, viruses, and components thereof), and may be used in identifying new medicines and therapeutic agents, and the uses thereof.No Admission of Prior Art

[0005] The discussion in this section is to provide background for the present disclosure and does not constitute an admission of prior art.SUMMARYHybrid Computing

[0006] The present disclosure provides for simulated evolution of a quantum circuit for hybrid computing of Hamiltonian eigensolutions. In a hybrid computing environment, both classical computing devices and quantum computing devices are used to provide operators with the benefits of each computing technology, while reducing the drawbacks of each computing technology. For example, by controlling the calculation of Hamiltonian eigensolutions in the hybrid environment as described in the present disclosure, an operator can produce a more accurate set of answers than when using a quantum-only computing system, and more quickly (and with fewer computing resources) than when using a cl as si cal -only computing system. Additionally, the added functionalities offered by the present disclosure include improvements over conventional hybrid computing systems by identifying when to perform given operations in a classical or a quantum computing system.Handover Algorithms

[0007] Traditional quantum computing practices take advantage of the increased efficiency of quantum computers to potentially solve various problems in fields of chemical analysis that have proven to be intractable for classical computers to solve. The classical computer traditionally prepares the problem to be run on the quantum computer and then decodes the output of the quantum computation to obtain the answer. However, quantum computers are still a developing technology for which hardware access remains limited, and the outputs provided by quantum computers can be inexact due to noise inherent in the quantum computers. Handover algorithms, as described in the present disclosure, allow the classical computer to take over calculations from the quantum computer before the quantum computer would traditionally output a solution but after the quantum computer has rendered the problemtractable by classical computers, which are not affected by quantum noise in their output and are generally available more widely than quantum computers. By identifying when the quantum computer has produced sufficient information for the quantum computer to “hand over” the task of completing the calculation to the classical computer, operators may perform calculations that that have traditionally been identified as intractable for classical computers while also avoiding noise in the final answer (producing more accurate final results), and freeing up limited quantum computing resources for other calculations that remain intractable for classical computers, among other benefits.Hamiltonian Calculations

[0008] Particularly, with respect to calculating Hamiltonian eigensolutions, by using the classical computing systems to formulate the initial full Hamiltonian, the quantum computing systems to produce the corresponding wave functions, and the classical computing system to determine how and when to produce a version of the full Hamiltonian with a reduced dimension (referred to herein as a subspace Hamiltonian), the present disclosure allows for control of the calculations to be passed back to the classical computing system. By controlling when and how to pass control back to the classical computing systems (or to pass operation back to the quantum computing systems), the present disclosure provides an output for Hamiltonian calculations that mitigates the uncertainty inherent to quantum noise in quantum computing systems. Additionally, the described approach uses fewer computing resources than other hybrid computing approaches and provides operators with various controls to select different thresholds for accuracy, resource dedication, and calculation profiles, among other benefits and technical improvements that will be apparent to those skilled in the art on a detailed review of the present disclosure.Hamiltonian Representation

[0009] The Hamiltonian can be represented as a matrix, which reduces the quantum chemistry problem to a matrix eigensolution problem. The size of such a matrix grows exponentially fast with respect to the number of orbitals. The target state that satisfies the eigensolution problem is often very sparse, meaning that the amplitudes, or contributions, of a large portion of the basis states that make up the state are zero or negligibly small. As a result, one can use the basis states that have non-negligible contributions to the target state to perform a reduction of the Hamiltonian matrix to a smaller dense matrix. Such a compressed matrix may still have an accurate eigensolution or have an approximated eigensolution.Extension of Solution

[0010] The eigensolution problem of a sparse matrix can be extended beyond chemistry 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 picture, the power of solving a large matrix can be derived from the efficient methodology for compressing a large matrix to a smaller representative matrix.Krylov Subspace

[0011] A Krylov subspace is a concept from linear-algebra, which can be used to form accurate approximate solutions to high dimensional problems. Various iterative methods (also referred to as Krylov subspace methods) may be used to find one or more eigenvalues for large sparse matrices, which generally avoid matrix-to-matrix operations in favor of the less computationally complex matrix-to-vector operations, and can even be used in situations without an explicitly defined matrix. Such methods may be represented in the iterative form Kr(A, b) = span {A°b, Axb, A2b, ... A'-1b } for an / '-order Krylov operation with respect to an n X n Matrix A and an / / -dimension Vector b.Aspects of the Disclosure

[0012] One aspect of the present disclosure provides a method that prepares basis states selected from a Krylov subspace of the full Hamiltonian developed on a quantum device using a quantum circuit that takes its structure from the Hamiltonian of the system under analysis to simulate the evolution of the Hamiltonian of that system. This set of states is sampled from to obtain the basis states used to construct a subspace Hamiltonian for further analysis. This method produces high quality basis states in the sampling step because the set of states prepared have a special quality, in that the states belong to a Krylov subspace.Variation is Contemplated across Iterations

[0013] In the forgoing method, the Krylov Handover procedure may include modification to one or more of the Hamiltonian, the states used in the Krylov subspace or the Hamiltonian, how feedback is handled in the system, the time-step parameters for evolution, the number of samples performed, and the like. Additionally, the initial state preparation protocol may comprise various different protocols that are varied across different iterations, the trial wave function preparation protocol may comprise various different protocols that are varied across different iterations, different stop criteria may be used across the various iterative loops and cause an individual loop to exit by different conditions at various iterations, the selectionprotocol for selecting the basis states for inclusion in the subspace Hamiltonian may comprise various different protocols that are varied across different iterations.Additional Aspects

[0014] Additional aspects, features and advantages of the disclosed method and apparatus are described in, and will be apparent from, the following Detailed Description and the Figures. The features and advantages described herein are not all-inclusive and, in particular, many additional features and advantages will be apparent to one of ordinary skill in the art in view of the figures and description. Moreover, it should be noted that the language used in the specification has been principally selected for readability and instructional purposes, and not to limit the scope of the inventive subject matter.BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 illustrates an example hybrid computing system, according to embodiments of the present disclosure.

[0016] Figures 2A-2E provide illustrations of various calculation spaces, according to embodiments of the present disclosure.

[0017] Figure 3 is an example block diagram of initial state feedback in Krylov Handover across iterations, according to embodiments of the present disclosure.

[0018] Figures 4A-4J are flowcharts of selection protocols for choosing which basis states from a full Hamiltonian to include in a subspace Hamiltonian, according to embodiments of the present disclosure.

[0019] Figures 5A-5J illustrate example graphs of performing certain selection protocols, according to embodiments of the present disclosure.

[0020] Figure 6 is a flowchart of an example method for performing a Krylov Handover when determining the ground state value and ground state energy of a Hamiltonian, as may be used to represent a chemical system, according to embodiments of the present disclosure.

[0021] Figures 7A-7C illustrate analysis of various chemical systems according to a Krylov handover process as described herein.

[0022] Figure 8 illustrates a classical computing device, according to embodiments of the present disclosure.

[0023] Figure 9 illustrates a quantum computing device, according to embodiments of the present disclosure.

[0024] Figures 10A-10S are slides describing aspects and embodiments of the present disclosure and the related fields of the art.

[0025] The exemplifications set out herein illustrate certain non-limiting embodiments, in one form, and such exemplifications are not to be construed as limiting the scope of the appended claims in any manner.DETAILED DESCRIPTIONExamples and Embodiments

[0026] The presently disclosed subject matter now will be described and discussed in more detail in terms of some specific embodiments and examples with reference to the accompanying drawings, in which some, but not all embodiments of the invention are shown. Like numbers refer to like elements or parts throughout unless otherwise referenced. The presently disclosed subject matter 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 so that this disclosure will satisfy applicable legal requirements. Indeed, many modifications and other embodiments of the presently disclosed subject matter will come to the mind of one skilled in the art to which the presently disclosed subject matter pertains. Therefore, it is to be understood that the presently disclosed subject matter is not to be limited to the specific embodiments disclosed and that modifications and other embodiments are intended to be included within the scope of the appended claims.Chemical Systems

[0027] The term “chemical system” may refer to any composition or collection of atoms representable by a chemical formula including molecules, ions, radicals, etc., in which the atoms thereof are bound together via covalent or non-covalent bonds, including homonuclear or heteronuclear collections thereof.Quantum Representation of a Chemical System

[0028] When representing a chemical system in a quantum computer, qubits are used to represent properties of chemical system, where each qubit is based on a two-level quantum system that encodes a computational basis state of |0) or |1), but can be a coherent superposition of either value at the same time until measured. Accordingly, a quantum computer using n qubits has 2nbasis states available to describe the chemical system. Each of these 2nbasis states is associated with a probability amplitude ca, whose magnitude squared represents a probability of the chemical system being in a given basis state, where the value of oa is determined by sampling the quantum representation of the chemical system and noting how often the chemical system is found in a given basis state. For example, if a one qubit system (e.g., n=l) is sampled X times, the computational basis state maybe determined to be |0) Y% of the time (e.g., Y = |a0|2* 100) and |1) Z% of the time (e.g., Z = | CI-L |2* 100), and the system can be presented in a wave function as |y) = a0|0> + c^ll).Hamiltonian (H)

[0029] A Hamiltonian is an operator in quantum mechanics, and for a chemical system represents the total energy and provides the possible energies according to a set of eigenvalues and eigenvectors. The energy for a given state in which the system can be is given by the expectation value of the Hamiltonian with respect to that state.Varieties of Hamiltonians

[0030] A full Hamiltonian includes values for every basis state for the chemical system it represents; however, many of these basis states’ contributions to the eigensolution are zero or close to zero, which indicates a low probability of the chemical system being measured in that basis state. Accordingly, the number of basis states that the chemical system has a high probability of being in are relatively small compared to the total number of basis states. Using this sparsity, a subspace Hamiltonian can be constructed using only the small number of basis states relevant to the eigensolution of the full Hamiltonian. If the basis states used to construct the subspace Hamiltonian match those that are most representative of eigensolution of the full Hamiltonian for the chemical system, then the eigensolution of the subspace Hamiltonian is able to approximate the eigensolution of the full Hamiltonian for the chemical system, while reducing the computational complexity of the representation.Krylov subspace

[0031] A Krylov subspace is a concept from linear-algebra, which can be used to form accurate (albeit) approximate solutions to high dimensional problems. Various iterative methods (also referred to as Krylov subspace methods) may be used to find one or more eigenvalues for large sparse matrices, which generally avoid matrix-to-matrix operations in favor of the less computationally complex matrix-to-vector operations, and can even be used in situations without an explicitly defined matrix. Such methods may be represented in the iterative form K,(A, b) = span {A°b, A'b, A2b, ... A'-Ib } for an / -order Krylov operation with respect to an n X n Matrix A and an / / -dimension Vector b.Implications of Identifying a Representative subspace Hamiltonian

[0032] Calculating the amplitudes for the basis states of the eigensolution of the full Hamiltonian for a chemical system is typically proposed to be performed via a quantum computer as the computational complexity of the calculations for chemical systems of increasing size renders generating a solution via classical computers or human mental processes intractable. Operators may use various quantum computing methodologies, such as a variational quantum eigensolver (VQE), to find the ground state of the chemical system from an initial state using an ansatz (which may develop a best guess for the eventual ground state formed using various methodologies known to those of skill in the art). VQE is an iterative process, which consumes significant computing resources on the quantum computer, and may end with the iterations resulting in suboptimal solution that does not correspond to the ground state (e.g., a local minimum instead of a global minimum), which further wastes limited quantum computing resources. However, by identifying when the quantum computer has produced a sufficiently accurate wave function to produce a representative subspace Hamiltonian (typically prior to reaching a quantitatively complete answer on its own), the quantum computer can hand over further calculations to a classical computer to solve the remaining portion of the calculations provided that the subspace Hamiltonian reduces the problem space enough to render the calculation classically tractable. Accordingly, a hybrid computing system can use a hand over procedure to transfer calculations (using a full Hamiltonian) from the quantum computer to a classical computing system (using a corresponding subspace Hamiltonian) to complete the calculations, thereby conserving limited quantum computing resources while also producing an output that is unaffected by a quantum noise floor for the accuracy thereof, among other benefits of using a hybrid computing system.Hamiltonian Reduction

[0033] Handover methodologies reduce the computational complexity of the eigensolution calculations so that a classical computer can return the final result without sacrificing accuracy, by relying on the generally sparse nature of the full space to be reduced to the subspace Hamiltonian. Stated differently, the computational complexity of the eigensolution calculation for these systems can only be reduced because the number of basis states required to form the eigensolution is extremely small compared to the total number of basis states in the search space of the system. By considering the small set of important basis states, rather than the full space of every basis state, the Hamiltonian can be greatly reduced in size and complexity; however, this set of important basis states is not known a priori, and computing resources must be spent to search for the appropriate subspace. The scaling of the search space for these systems, however, can render any such methods computationally infeasible.Trial State Sampling and Evolution

[0034] Rather than searching for an optimal subspace, by iteratively updating the quantum circuit to obtain a state that better overlaps with the most significant basis states of the actual target state, the present disclosure contemplates preparing a state that yields representative sample basis states, but does not itself form an approximation of the target state. Accordingly, a set of states are prepared on a quantum device using a known quantum circuit that simulates the evolution over time of the Hamiltonian of the system. A subspace Hamiltonian is constructed with basis states obtained by sampling from these states, which produces high quality basis states when the set is a Krylov subspace.Circuit Preparation

[0035] The power of the Krylov states arises from preparation using quantum circuits which implement a function of the Hamiltonian of the chemical system. For example, for a function of (x) = Hx, then a Krylov subspace may be created forstates as KSfc= { / (O)lv>o>, / (l)lv>oh ->f(Sk- Wo» = . Another possible function to use is the real-time evolution of the Hamiltonian, which produces good basis states when performing the sampling, even though the energy of the state on the quantum device remains unchanged throughout the circuit execution. Additionally, the simulation of Hamiltonian evolution is one of the applications for which quantum computers will offer an exponential speedup compared to classical computers. When using a real-time evolution, the function for the Krylov subspace can be defined as (x) = e~lHxt, where the parameter t specifies the time of evolution.Iterative Behavior

[0036] In the definition of the Krylov states, the term | / o) is the initial state prepared on the quantum device. For chemical systems, this can be the Hartree-Fock state or any other quantum state which has a good overlap with the target state. The quality of the Krylov subspace depends on, among other factors, the dimension of the subspace (how many quantum states the subspace comprises), the time step used in the circuit, and the initial state. By updating / changing any of these factors from one iteration to the next, the Krylov subspace may iteratively evolve. In some embodiments, the eigensolution from the previous iteration is used as the initial state for the next iteration on the quantum device, thereby offering increased overlap with the target state.Inter-relation of Spaces and States

[0037] As will be appreciated, a Krylov state is a quantum state prepared using a circuit that implements some function of the Hamiltonian, applied to an initial state, and a Krylov subspace is a set of Krylov states. Basis states, which are computational representations of the quantum probabilities of a given state, can be sampled from the Krylov states to construct a subspace of the Hamiltonian representation of a chemical system.Orthonormality

[0038] Sampling from the Krylov states to obtain basis states which are used to create the subspace Hamiltonian offers several benefits over the conventional procedure of representing the Hamiltonian using the quantum states in the Krylov subspace, as these conventionally represented states are not orthonormal (meaning that these states have some overlaps that are non-zero). In linear algebra, two (or more) vectors in a product space are determined to be orthonormal when those vectors are perpendicular to one another (e.g., orthogonal) and of a unit length. As a result of not being orthonormal, the overlaps of these quantum states must be calculated and taken into account when solving the eigensolutions of the Hamiltonian. Calculating these overlaps must be performed on the quantum computing device, and dramatically increases the quantum resources required by the algorithm. The present disclosure, by instead sampling orthonormal basis states in the computational basis from the quantum states in the Krylov subspace, ensures that all of these basis states have mutual overlaps of zero (because these sampled states are orthonormal), and thus does not incur the extra quantum resource overhead present in conventional methodologies.Contemplated Variations

[0039] Additionally or alternatively to the usual control possibilities that the user is afforded in conventional Krylov analyses and in iterative sampling methodologies, the present disclosure contemplates that additional variations that can be introduced into the presently described Krylov Handover procedure.Hamiltonian Modifications

[0040] In various embodiments, control possibilities for the described Krylov Handover procedure can include various rules-based modifications to the chemical Hamiltonian used to implement the quantum circuit. For example, a truncation rule can be applied wherein Pauli terms in the Hamiltonian are removed if the absolute value of the coefficients of those Pauli terms is below some defined threshold, which would reduce the number of Pauli terms, thereby reducing the depth of the quantum circuit.Initial State Modifications

[0041] In various embodiments, control possibilities for the described Krylov Handover procedure can include various modifications to the initial states for the states in the Krylov subspace. For example, an operator may update the initial state every iteration, update the initial state for some subset of states in the Krylov subspace every iteration, use a different initial state for each state in the Krylov subspace, etc. The chosen initial state modification scheme can change the characteristics of the states in the Krylov subspace, which can increase the efficiency with which the system is able to sample important basis states therefrom.Time-Step Parameter in Quantum Circuit

[0042] In various embodiments, when using the real-time evolution of the chemical Hamiltonian as the quantum circuit to generate the states in the Krylov subspace, control possibilities for the described Krylov Handover procedure can include various values or formulas selected for defining the time term t. Because the quality of the basis states that can be sampled is dependent on the parameter t used in the circuit, which specifies the time step for the evolution, various (non-exhaustive) options for varying t may include: using a different time step for each state in the Krylov subspace, using the same time step for each state in the Krylov subspace, selecting the time-step according to some a priori knowledge about the chemical system, optimizing the time-step over several iterations of the Krylov Handover method.Operator generation of Krylov subspace

[0043] In various embodiments, although a real-time evolution operator may be used to generate the states in the Krylov subspace, one can use any function of the Hamiltonian of the chemical system. For example, an operator may select (as an alternative or parallel analysis to the real-time evolution operator) to use powers of the Hamiltonian itself, use the real-time evolution of the Hamiltonian according to some time-steps, or another formula or operator for some or all of the analysis.Number of Samples

[0044] In various embodiments, control possibilities for the described Krylov Handover procedure can include adjustments to the number of samples performed on the quantum device. Because the number of samples taken forms a major factor in the determination of the quantum resources required to perform the algorithm, allocating the number of samples can have significant effect on the efficiency of the operations. Some examples for sample number adjustment include: performing the same number of samples from each quantum state in the Krylov subspace, distributing the number of samples between the quantum states in the Krylov subspace according to some rule, (e.g., performing more samples on the states that were evolved for longer), random sample numbering, or the like.Hybrid Computing System

[0045] Figure 1 illustrates an example hybrid computing system 100, according to embodiments of the present disclosure. The hybrid computing system 100 includes a classical computing system 110 that is interfaced with a quantum computing system 120 to perform calculations partially using classical computing techniques and quantum computing techniques. As used herein, a hybrid computing system 100 refers to a classical-quantum computer combination, rather than combinations of analog and digital computers, which in other fields have variously referred to as “hybrid computers”. Each of the classical computing system 110 and the quantum computing system 120 may include one or multiple computing devices and various communications interfaces. An example classical computing system 110 is discussed in relation to Figure 8, and an example quantum computing system 120 is discussed in relation to Figure 9.Benefits of a Hybrid Computing System

[0046] Conventionally, classical computing systems 110 have been used to define the sequence of quantum gates or quantum circuits for the quantum computing systems 120 and to receive the results of the quantum calculations performed by quantum computing systems 120.In contrast, the hybrid computing system 100 allows for both classical calculations and quantum calculations to be efficiently performed by distributing the workload between the systems according to the relative strengths of the systems for certain tasks. This results in improving the functionality of the overall system, reducing the computational resources required by conventional computing systems, improving the speed and accuracy of the calculations performed, among other technical improvements and benefits.Various Architectures of Hybrid Computing System

[0047] In various embodiments, the hybrid computing system 100 may be organized according to various architectures, including batch quantum computing architectures, interactive quantum computing architectures, integrated quantum computing architectures, and distributed quantum computing architectures.Batch Computing Architectures

[0048] In batch quantum computing architectures, the classical computing system 110 defines quantum circuits and submits those quantum circuits as jobs to the quantum processing unit (QPU) of the quantum computing system 120, which returns the result to the classical computing system 110. Batching multiple quantum circuits into one job, however, reduces wait time between submissions, allowing the hybrid computing system 100 to run multiple jobs faster. Accordingly, when one quantum circuit “job” is completed by the quantum computing system 120, the next quantum circuit “job” is ready for analysis.Interactive Quantum Computing Architectures

[0049] In interactive quantum computing architectures, an operator can specify repeated execution of the quantum circuit with different parameters (or the same parameters). Jobs can be grouped logically into one session and prioritized over non-session jobs. Although sessions allow for shorter queue times and longer running problems, the qubit states does not persist between each iteration. Examples of problems that can use this approach are Variational Quantum Eigensolvers (VQE) and Quantum Approximate Optimization Algorithms (QAOA).Integrated quantum computing architectures

[0050] In integrated quantum computing architectures, classical computations are performed while physical qubits are coherent, as the classical computing system 110 and quantum computing system 120 operate together. Although potentially constrained by qubit life and error correction, integrated architectures allow for quantum programs to include commonprogramming tools (e.g., loops, nested conditional statements, etc.) beyond mere circuit analysis that can use one or more states from the quantum circuit as values for variables. Beneficially, the classical computing system 110 may allow for various qubit reuse techniques; allowing the quantum computing system 120 to have fewer physical qubits, but offer a larger number of “virtual” qubits to run more complex calculations.Distributed Quantum Computing Architectures

[0051] In distributed quantum computing architectures, the classical computing system 110 works with the quantum computing system 120 using logical qubits. The long coherence time provided by logical qubits enables complex and distributed computation across a cloud computing environment. Accordingly, the various computing devices may be located in different environments from one another, and may be shared with other users on a platform as a service (PaaS) usage model.Example Calculation Spaces

[0052] Figures 2A-2E provide illustrations of various calculation spaces, according to embodiments of the present disclosure. Figures 2A-2B illustrate representations of an electron Hamiltonian for a hydrogen molecule (H2), in which Figure 2A is a matrix representation of the full Hamiltonian of H2, and Figure 2B illustrates an example matrix representation of a subspace Hamiltonian of H2. Figure 2C illustrates a Pauli representation for the Hamiltonian of H2. Figure 2D illustrates a circuit block for the exponential of a single Pauli term in the Hamiltonian of H2, and Figure 2E illustrates the quantum circuit for the Hartree-Fock initial state of H2.Hamiltonian Matrix

[0053] A Hamiltonian can be represented with a matrix such that when the quantum computing device uses n qubits on the quantum computer, the dimension of the Hamiltonian matrix can be described as 2nx 2n. For the H2 molecule which uses 4 qubits, the molecule can be fully represented in a 16 X 16 Hamiltonian matrix as in shown in Figure 2A.Hamiltonian Matrix for H2

[0054] The values shown in the matrix of Figure 2A represent the matrix elements for each of the different basis states, where each position ij in the matrix corresponds to the matrix element of the basis states btand bj. As will be apparent, the eigensolution of the full Hamiltonian can be well approximated by a sparse solution; many of the basis states haveamplitudes of zero or near-zero, indicating a zero or near-zero contribution to the eigensolution of the full Hamiltonian.Linear Sum of Pauli Words

[0055] As will be appreciated, this matrix can also be represented as a complex linear sum of the Pauli words, which is more memory efficient, as the complexity of the matrix grows exponentially relative to the number of qubits. As one of skill in the art will appreciate, the complex linear sum representation of the full Hamiltonian is what the VQE operates on due to the memory efficiency of this format, despite both encoding the same data, although various black-box representations with an element-wise access protocol may also be used for even further increases in efficiency.Subspace Hamiltonians

[0056] To reduce the computational complexity of the mathematical operations to perform on the full Hamiltonian, a subspace Hamiltonian is created. The subspace Hamiltonian is a reduced matrix whose eigensolution approximates that of the full Hamiltonian, allowing a classical computer to solve for an approximate eigensolution of the full Hamiltonian, a task usually considered intractable on classical devices due to the exponential size of the matrix. Due to the sparse nature of the approximation of the eigensolution of the full Hamiltonian, with many amplitudes of zero or near-zero, the basis states with zero or near-zero amplitudes have a negligible effect on the calculation of the approximate eigensolution of the full Hamiltonian. Accordingly, this subspace Hamiltonian provides a highly accurate approximation of the full Hamiltonian, such that the classically computed energy is also highly accurate, provided that the important basis states (those with non-negligible amplitudes in the eigensolution) are used to construct the subspace Hamiltonian.Evolution Development contrasted to Variational Circuit Training

[0057] In addition or alternatively to generating basis states using a parameterized quantum circuit for which the circuit parameters are iteratively updated to improve the overlap of the prepared quantum state with the target state, a Krylov handover operation may be used wherein the quantum circuit used implements an evolution of the initial state by the Hamiltonian. After creating the Hamiltonian (and potentially pre-processing the Hamiltonian to apply various filters or modifications), the Hamiltonian may be converted from a matrix representation (as in Figure 2A) to a Pauli-words representation (as in Figure 2C). The quantum circuit to prepare the Krylov subspace is developed from the Pauli-representation, and when sampled mayprovide basis states that cannot be efficiently obtained from an iteratively updated parameterized quantum circuit that aims to prepare an approximation of the target state. Stated differently, higher quality basis states may be yielded from the Krylov subspaces compared to basis sates obtained via a variational training approach when challenges exist for preparing known-good approximation to the target state.Krylov Space Development

[0058] The Pauli-representation of the Hamiltonian can be converted into a quantum circuit via the Trotter Decomposition process or a similar process, which yields a circuit block for the exponential of each Pauli term in the Hamiltonian, which is shown in Figure 2D. To generate the full quantum circuit, the circuit blocks for each Pauli term are chained together. The quantum circuit is applied to an initial state (which can be any valid state for the chemical system) to create a state in the Krylov subspace. For example, an eigensolution from a previous iteration may be used, or a Hartree-Fock state may be used. In case example for H2, when using 4 qubits, the circuit to create the Hartree-Fock state is shown in Figure 2E. Each state in the Krylov subspace uses the same circuit structure, but the parameters of various gates may differ according to the time of evolution to produce different output values from the circuit. For example, if the Krylov subspace were to comprise three states based on { / , 2 / , and 3t} as the times of evolution, then the quantum circuit would behave as three circuits. As will be appreciated t is a real-valued number (e.g., t = 0.01), and an operator may select various values for t in addition to the evolutions thereof. Accordingly, the Krylov subspace may be prepared as the three states of: e~lHt\ipHF), e~lH2t\ipHF), and e~lH3t\ipHF). However, as the size of the chemical system increases, the quantum circuit may use all of the available qubits, and the states may need to be prepared one at a time, although smaller systems may be prepared in parallel.Initial State Feedback Iterations

[0059] Figure 3 is an example block diagram of initial state feedback in Krylov Handover across iterations, according to embodiments of the present disclosure. Using the example Sk states 310a-Sk of e-t^t|? / ;7-1), e-t^2t|? / ;7-1), ... e-l^SKt|^7-1), with the iteration tracked via variable j, each time evolution can be used to generate sets of samples 320a-Sk to generate an eigenstate 330 for the iteration, which in turn is fed into the next iteration to use as the initiate state thereof for each of the states 310a-Sk. The eigenstate 330 can be prepared on the quantum device as an initial state using a polynomial number of quantum gates as the eigenstate 330 isa sparse state in the Hilbert space of the full Hamiltonian. As will be appreciated, preparation of an arbitrary quantum state requires an exponential number of quantum gates normally, as there are an exponential number of amplitudes that need to be set to recreate the state. However, when the eigenstate is sparse, as for a chemical system, the state can be prepared with a polynomial number of quantum gates, which can be performed in polynomial time.Krylov Sampling

[0060] The states prepared on the quantum device are sampled to obtain a subset of basis states that are orthonormal to each other. Sampling may be performed according to various methodologies used substantially identically to sampling performed from the Hamiltonian’s Hilbert space according to various selection protocols, where several measurements are performed on each qubit on the quantum device to obtain a binary string of ones and zeroes of equal length to the number of qubits to form the basis states. These basis states are used to construct a subspace Hamiltonian that may be analyzed to produce an eigensolution for the chemical system. Because the noise on the quantum device may yield basis states that are invalid or yield a different number of basis states for these samples than is desired, the sampled basis states may be filtered or supplemented to ensure that a desired number (K) of valid basis states are used for construction of the subspace Hamiltonian.Factors for Selecting K Value

[0061] Because operators do not have unlimited power to select the value of K for how many basis sates to use in computing an eigensolution, certain factors may be used to select an appropriate K value. These factors include, but are not limited to: (1) the probabilities developed for the quantum state being sampled, (2) the available computing resources for the quantum computer, (3) the available computing resources for the classical computer, and (4) a prior value for K in the case that the prior selected value was insufficient for calculating the eigensolutions (e.g., to ensure a larger value to K is selected on a subsequent iteration).Sampled Values

[0062] The first factor for consideration is what the amplitudes of the basis states in the quantum state being prepared are. The amplitudes (m) of the quantum state, give the probability | |2of sampling the z111basis state. For example, if an operator intends to sample desired states, but the number (j) of basis states in the quantum state that have non-negligible probabilities of being sampled is less than Xdesired (e.g., j < Xdesired), then the highest value for / factual is j or less (e.g., ^factual < j). This could be the case if the hand over from the quantum computer to theclassical computer were performed too soon and the amplitudes of the basis states in the quantum state were not qualitatively accurate enough before the state was sampled. In practice, the noise on the quantum computer will likely offset this effect, and allow the sampling of a sufficient number of basis states. This offset does not mean that the algorithm can be terminated earlier on a noisy device though, because there is no reason to believe that the basis states brought about by noise should have non-zero contributions in the final eigensolution.Available Quantum Computing Resources

[0063] The next factor is the quantum computing resources available, which includes the amount of samples that an operator can afford to take when sampling the basis states from the quantum state. Operators have some finite number of attempts in order to sample the basis states (e.g., based on time, processor cycles, time of holding a value in a qubit), and if the operator only samples some number j, where j < ^desired, of basis states within those samples, the operator will have no choice for the size of j = factual.Operator Input for K Selection

[0064] The above two factors do not necessarily limit the size of ^desired. For example, an operator can always guess samples that should be non-zero in the final solution, or provide values from external sources (or previous runs of the procedure). However, these two factors do present a hard limit on the number of basis states that can be sampled directly from one quantum state / iteration of the process as the quantum computing device (and access thereto) may only provide a given number of results.Available Classical Computing Resources

[0065] The third factor is the classical computing resources available. These have two restraints: the time and the space (memory) available to the classical computer. The value of K needs to be small enough that the classical computer has the resources required to compute the eigensolution of the subspace Hamiltonian. For example, if all possible basis states were to be used, because classical computing time and space required to find the eigensolution of a Hamiltonian grows exponentially with the problem size, the resources needed by the classical computer could make the calculations practically impossible on the most powerful classical computing systems for even moderately sized chemical systems.Prior Values for K

[0066] The fourth factor provides a counterbalance to pressure of the third factor to choosea lower value for Xdesired and the desire for more similar subspace Hamiltonians to the full Hamiltonians to choose a higher value for fdesn-ed. For example, if a value selected for K in an earlier iteration is computationally determined to be too low to create a representative subspace Hamiltonian, the system will ensure that the next iteration does not select the same or a lower value for K. In contrast, if a value selected for K in an earlier iteration remains computationally intractable for the classical computing system, the system will ensure that the next iteration does not select the same or a higher value for K. Accordingly, the hybrid system may initially attempt lower values for Xdesired in earlier iterations of a handover procedure, but continually increase the value of Xdesired in subsequent iterations when the evaluation of the produced subspace Hamiltonian is judged to not be representative of the full Hamiltonian. By implementing a rule according to the fourth factor, the hybrid system may gradually adjust the computational load that the classical computing system is assigned until a represented subspace Hamiltonian is produced. As will be appreciated, adjusting the size of the value for K can increase or decrease the number of basis states included in a subspace Hamiltonian compared to earlier iterations; allowing an operator to fine-tune the computational load and accuracy of the solution produced by classical computing system.Selection Protocols

[0067] In addition to the benefits provided by being able to adjust the number of basis states used to create a subspace Hamiltonian across several iterations (e.g., selecting different values for K in different iterations), the present disclosure provides several possibilities for selection protocols to determine which basis states are selected from the Hilbert space of the full Hamiltonian to be the K basis states used to construct a subspace Hamiltonian. Accordingly, by using a different selection protocol in different iterations, even if the same value for K is selected for use with the same full Hamiltonian, several different subspaces are developed, and similarly several different subspace Hamiltonians can be generated, which may prioritize different aspects for determining which basis states from the Hilbert space of the full Hamiltonian to include in an analysis subset as the K basis states for analysis.Projection of the Hamiltonian when Using Krylov Subspaces with Basis State Selection

[0068] In summary, basis states are selected by sampling from the quantum state. These basis states belong to the Hilbert space of the Hamiltonian, and the best basis states are selected from the Hilbert space according to one or more selection protocols. After selection, the selected basis states may be referred to as a subspace, which is used to project the full Hamiltonian intoa smaller matrix.Methods for Selecting Basis States from full Hamiltonians to include in Subspace Hamiltonians

[0069] Figures 4A-4J are flowcharts of selection protocols 400a-400j (generally or collectively, selection protocol 400) for choosing which basis states from the Hilbert space of the full Hamiltonian to use to construct a subspace Hamiltonian and Figures 5A-5H provide example graphs 505a-h (generally or collectively, graphs 505) of performing certain selection protocols, according to embodiments of the present disclosure. In the graphs 505, nodes 515a-t (generally or collectively, nodes 515) each represent one sampled basis state, and selected nodes 515 are represented by solid circles, while unselected nodes are represented by un-filled circles.Electron Preserving Selection Protocol

[0070] Figure 4A is a flowchart of a method 400a for implementing an electron preserving selection protocol, according to embodiments of the present disclosure. The electron preserving selection protocol selects the K basis states from the A basis states sampled from the quantum device (e.g., as a subset of the available basis states, where K < A), where the bitstring representing each selected basis state in the set of K contains the same number of 1’s as there are electrons being considered in the problem. Stated differently, method 400a applies a symmetry criterion for the selection of sample basis states to include from the Hilbert space of the full Hamiltonian, to use to construct a subspace Hamiltonian, in which the symmetry criterion is satisfied based on selecting the number of values of the second plurality of values equaling K that are represented by bitstrings having a number of l’s equaling a number of electrons in a chemical system represented by the full Hamiltonian.Block 410a Sampling N Basis States from the Quantum Device

[0071] At block 410a, the quantum computing system samples a trial wave function to obtain the set of A basis states from among the 2nbasis states, with each basis state b, having a probability | |2of being sampled.Block 420a - Removing Bitstrings Not Containing Corresponding Numbers of l ’s

[0072] At block 420a, the classical computing system removes the basis states from consideration for inclusion in the construction of the subspace Hamiltonian that do not have bitstrings that contain a number of l’s that match the number of electrons in the chemicalsystem under analysis. As used herein the number of bitstrings determined to have the same number of l’s as the number of electrons is referred to as EBS. For example, in a chemical system with two electrons, the basis state with bitstring '1010' would be selected amongst the N sampled basis states as the number of l’s is equal to the number of electrons; however the basis state with bitstring '0111' would not be selected as that bitstring has three 1 ’ s, while there are two electrons in the problem. Each of the bitstrings with the corresponding number of l’s would be available for selection as part of the K selected basis states, and the value for EBS would be six (e.g., 0011, 0101, 0110, 1001, 1010, and 1100 for two electrons). Accordingly, of the sixteen bitstrings (e.g., 0000 to 1111) sampled for an exhaustive sampling of a system with two electrons and four qubits, ten would be removed; leaving six for selection as part of K (e.g., EBS = 6).Block 430a - Supplemental Selection / Removal

[0073] At block 430a, basis states for selection in calculating the eigensolution are chosen from the EBS basis states identified as satisfying the selection protocol. When Ess= i (or when EBs= remainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when EBS > K, a supplemental selection process may be performed to identify a subset of the EBS basis states to reduce the selection to K basis states for use in the subspace Hamiltonian. Similarly, when EBS < K, a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in the construction of the subspace Hamiltonian (e.g., ^remainder = desired - EBS). Generally, to ensure that a set of desired basis states is eventually selected, one or more sets of ^supplemental basis states may be selected to add to an initial amount of Aiected basis states (where the individual basis states of the Supplemental basis states preferably are not members of the initial Selected basis states) or one or more sets of Kremovai basis states may be selected from the Selected basis states to remove members from the initial amount of Selected basis states. Once the esired basis states are determined for constructing the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Alpha-Beta Preserving Selection Protocol

[0074] Figure 4B is a flowchart of a method 400b for implementing an alpha-beta electron preserving selection protocol, according to embodiments of the present disclosure. The alphabeta preserving selection protocol selects the K basis states from the N basis states sampledfrom the quantum device (e.g., as a subset of the available basis states, where K < TV), where the bitstring representing each selected basis state in K contains the same number of l's in the alpha orbital positions as there are alpha electrons in the problem, and the bitstring representing each selected basis state in K contains the same number of l's in the beta orbital positions as there are beta electrons in the problem. Stated differently, method 400b applies a symmetry criterion for the selection of sample basis states to include from the Hilbert space of the full Hamiltonian, to use to construct a subspace Hamiltonian, in which the symmetry criterion is satisfied based on selecting the number of values of the second plurality of values equaling K that are represented by bitstrings having a number of l’s in positions corresponding to a given type of orbital equaling a number of electrons in the given type of orbital of a chemical system represented by the full Hamiltonian.Block 410b Sampling N Basis States from the Quantum Device

[0075] At block 410b, the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state b, having a probability | |2of being sampled.Block 420b - Removing Bitstrings Not Containing Corresponding Numbers of l ’s in Corresponding Orbitals

[0076] At block 420b, the classical computing system removes the basis states from consideration for inclusion in construction of the subspace Hamiltonian that do not have bitstrings that contain a number of l’s that match the number of electrons in the positions corresponding to the alpha and beta orbitals in the chemical system under analysis. As used herein, the number of electrons in the alpha orbitals is identified as Eaand the number of electrons in the beta orbitals is identified as Ep. For example, for a problem with two alpha electrons (e.g., Ea=2), 4 alpha orbitals, two beta electrons (e.g., Ep=2), and 4 beta orbitals the chemical system can be represented with an eight-bit bitstring with the first half of the bits in the bitstring corresponding to the alpha orbitals and the second half of the bits in the bitstring corresponding to the beta orbitals (e.g., as aaaaPPPP), although one of skill in the art will appreciate that other representations can be used. Continuing the example, the classical computing system would remove any basis state with a bitstring with fewer than two l’s or more than two l’s in either the alpha portion or the beta portion (e.g., 11101100, which has three l’s in the alpha portion and two Is in the beta portion; 10101000, which has two l’s in the alpha portion and one 1 in the beta portion, etc.). In contrast, the classical computing systemretains the basis states that have two l’s in each of the alpha and beta portions (e.g., 11001010 and 10101001) for use in the K selected basis states. The total number of basis states that have the bitstrings that satisfy both Eaand Ep may be understood as EBS.Block 430b - Supplemental Selection / Removal

[0077] At block 430b, basis states for selection in calculating the eigensolution are chosen from the EBS basis states identified as satisfying the selection protocol. When Ess= i (or when EBs= remainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when EBS > K, a supplemental selection process may be performed to identify a subset of the EBS basis states to reduce the selection to K basis states for use in the subspace Hamiltonian. Similarly, when EBS < K, a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in constructing the subspace Hamiltonian (e.g., ^remainder = desired - EBS). Once the K basis states are determined for construction of the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Overlap Partition Selection Protocol with Partition Bias

[0078] Figure 4C is a flowchart of a method 400c for implementing an overlap partition selection protocol with a partition bias, according to embodiments of the present disclosure. The overlap partition selection protocol selects from the N basis states sampled from the quantum device a set of K basis states (K < N), where each selected basis state in K is in the same graph partition as a known reference state. Stated differently, method 400c applies an overlap criterion for the selection of sample basis states to include from the Hilbert space of the full Hamiltonian, to construct the subspace Hamiltonian, in which the overlap criterion is satisfied based on generating a graph of the first plurality of values as a plurality of nodes with edges connecting every pair of nodes for which <bi|H|b2> is non-zero; identifying a reference node from the plurality of nodes associated with a known reference state; and selecting all nodes of the plurality of nodes that share a partition in the graph with the reference node.Block 41 c Sampling N Basis States from the Quantum Device

[0079] At block 410c, the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state b, having a probability | |2of being sampled.Block 420c - Graph Construction

[0080] At block 420c, the classical computing system constructs a graph (such as graph 505a shown in Figure 5 A) by creating a node / vertex for each of the basis states in the set of A basis states sampled from the quantum device. Each basis state then has a corresponding node / vertex in the graph.Block 430c - Add Edges

[0081] At block 430c, the classical computing system adds an edge for every combination of two nodes / vertices in the graph, representing two basis states |bi> and |b2>, where <bi|H|b2> is non-zero. Accordingly, as shown in graph 505a in Figure 5A, edges are constructed between some of the nodes / vertices, and some sets of nodes / vertices (e.g., a graph partition) may be disconnected from other sets of nodes / vertices (e.g., other graph partitions).Block 440c - Retain Nodes In Shared Graph Partition with Reference Node

[0082] At block 440c, the classical computing system retains the basis states associated with a known reference state in a shared partition of the graph. Any two basis states are in the same partition if and only if their corresponding nodes / vertices are connected to each other by a path. For example, in Figure 5 A, if node 515a is identified as the reference state, the other nodes 515b-l in the same partition with the node 515a are retained, while nodes 515m-o and 515p-t are removed from consideration. As used herein, the number of basis states retained in the shared partition is denoted as Pnodes.Block 450c - Supplemental Selection / Removal

[0083] At block 450c, the basis states for selection are chosen from the basis states identified as belonging to the same graph partition as the reference state. When Pnodes=A (or when Pnodes=Xremainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when Pnodes > K, a supplemental selection process may be performed to identify a subset of the Pnodes basis states to reduce the selection to K basis states for use in the construction of the subspace Hamiltonian. Similarly, when Pnodes < K, a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in constructing the subspace Hamiltonian (e.g., ^remainder = Aiesn-ed - Pnodes). Once the K basis states are determined for construction of the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Overlap Partition Selection Protocol with Breadth-First Bias

[0084] Figure 4D is a flowchart of a method 400d for implementing an overlap partition selection protocol with breadth-first bias, according to embodiments of the present disclosure. Nodes within a lower number of “hops” (having to traverse a smaller number of edges) from the reference node are given priority for inclusion in the construction of the subspace Hamiltonian over nodes further from the reference node or in different partitions. Stated differently, method 400d applies an overlap criterion for the selection of sample basis states to include from the Hilbert space of the full Hamiltonian, to construct the subspace Hamiltonian, in which the overlap criterion is satisfied based on generating a graph of the first plurality of values as a plurality of nodes with edges connecting every pair of nodes for which <bi |H|b2> is non-zero; identifying a reference node from the plurality of nodes associated with a known reference state; and selecting all nodes one edge away from the reference node or a previously selected node from the plurality of nodes until all nodes of the plurality of nodes that share a partition with the reference node are selected or at least K nodes from the plurality of nodes are selected, whichever occurs first.Block 410d Sampling N Basis States from the Quantum Device

[0085] At block 41 Od, the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state bj having a probability | |2of being sampled.Block 420d Graph Construction

[0086] At block 420d, the classical computing system constructs a graph (such as graph 505b shown in Figure 5B) by creating a node / vertex for each of the basis states in the set of A basis states sampled from the quantum device. Each basis state then has a corresponding node / vertex in the graph.Block 430d- Add Edges

[0087] At block 43 Od, the classical computing system adds an edge for every combination of two nodes / vertices in the graph, representing two basis states |bi> and |b2>, where <bi|H|b2> is non-zero. Accordingly, as shown in graph 505b in Figure 5B, edges are constructed between some of the nodes / vertices, and some sets of nodes / vertices (e.g., a graph partition) may be disconnected from other sets of nodes / vertices (e.g., other graph partitions).Block 440d Breadth-First Expansion

[0088] At block 440d, the classical computing system identifies which of the basis states to retain based on a breadth-first tree expansion starting with the node / vertex of a known reference state. This breadth-first expansion expands from the reference state node / vertex to first include basis states whose node / vertex is connected to the reference state node / vertex in the graph, as is shown in graph 505c in Figure 5C with the selection of nodes 515b, 515c, and 515i compared to graph 505b in Figure 5B. Next, basis states whose node / vertex is connected to nodes / vertices that are connected to the reference state node / vertex in the graph are included, such as is shown in graph 505d (e.g., via nodes 515d, 515g, 515j and 5151) in Figure 5D, and so on. This expansion continues until either the nodes / vertices of K basis states have been included in the selection, or there are no more basis states with nodes / vertices in the graph partition containing the reference state node / vertex to include. As used herein, the number of basis states retained in the partition from the breadth-first expansion is denoted as Pnodes.Block 450d Supplemental Selection

[0089] At block 450d, up to K basis states for selection are chosen from the basis states identified as being within a breadth-first expansion from the reference node (e.g., expansion per block 440d may conclude once the nodes for K basis states are identified). When Pnodes= i (or when Pnodes= iremainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when Pnodes < K, a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in constructing the subspace Hamiltonian (e.g., Remainder = idesired - Pnodes). Once the K basis states are determined for construction of the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Overlap Partition Selection Protocol with Best-First Bias

[0090] Figure 4E is a flowchart of a method 400e for implementing an overlap partition selection protocol with best-first bias, according to embodiments of the present disclosure. Stated differently, method 400e applies an overlap criterion for the selection of sample basis states to include from the Hilbert space of the full Hamiltonian, to construct the subspace Hamiltonian, in which the overlap criterion is satisfied based on generating a graph of the first plurality of values as a plurality of nodes with edges connecting every pair of nodes for which (bi|H|b2> is non-zero; assigning a score to each node of the plurality of nodes based on a heuristic measure of how important a corresponding basis state is; identifying a reference nodefrom the plurality of nodes associated with a known reference state; and selecting a next node one edge away from the reference node or a previously selected node from the plurality of nodes that has the highest score relative to all other nodes of the plurality of nodes that are one edge away from the references node or any previously selected node from the plurality of nodes until all nodes of the plurality of nodes that share a partition with the reference node are selected or at least K nodes from the plurality of nodes are selected, whichever occurs first.Block 410e Sampling N Basis States from the Quantum Device

[0091] At block 41 Oe, the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state bj having a probability | |2of being sampled.Block 420e - Graph Construction

[0092] At block 420e, the classical computing system constructs a graph (such as graph 505e shown in Figure 5E) by creating a node / vertex for each of the basis states in the set of A basis states sampled from the quantum device. Each basis state then has a corresponding node / vertex in the graph.Block 430e Add Edges

[0093] At block 43 Oe, the classical computing system adds an edge for every combination of two nodes / vertices in the graph, representing two basis states |bi> and |b2>, where <bi|H|b2> is non-zero. Accordingly, as shown in graph 505e in Figure 5E, edges are constructed between some of the nodes / vertices, and some sets of nodes / vertices (e.g., a graph partition) may be disconnected from other sets of nodes / vertices (e.g., other graph partitions).Block 440e - Assign Heuristic Score

[0094] At block 440e, the classical computing system assigns each node in the graph a score based on a heuristic measure, such as how relevant the corresponding basis state is, a relative importance of a given basis state, etc.. In various embodiments, the heuristic score is calculated by constructing the overlap of the basis state bi with the reference node or another currently selected node.Block 450e - Best-First Expansion

[0095] At block 450e, the classical computing system identifies which of the basis states to retain based on a best-first tree expansion starting with the node / vertex of a known referencestate. This expansion then selects the next best scoring node / vertex that is allowed to be expanded within the partition to which the reference node belongs to while growing the tree structure. In various embodiments, the score of each unexpanded node is reassigned by the classical computing system after the expansion of the current node. The tree structure grows in this manner until either K nodes / vertices of are expanded into or all of the nodes within the given partition are expanded into. As used herein, the number of basis states retained in the partition from the best-first expansion is denoted as Pnodes.Example Best-First Expansion

[0096] As shown in the graph 505e in Figure 5E, the initial selection for the basis states to include starts with the reference node 515a. The expansion proceeds to graph 505f in Figure 5F by selecting the second node 515b based on the second node 515b having the highest score for any node 515 linked via an edge to any of the currently selected nodes (note that node 515h has a higher score than node 515b, but is not selected because node 515h is unconnected by an edge to any currently selected node 515). Expansion may continue as shown in graph 505g and graph 505h by selecting the highest-scoring nodes connected by an edge to one of the currently selected nodes.Block 460e - Supplemental Selection

[0097] At block 460e, up to K basis states for selection are chosen from the basis states identified as being within a best-first expansion from the reference node (e.g., expansion per block 450e may conclude once the nodes for K basis states are identified). When Pnodes= i (or when Pnodes= iremainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when Pnodes < K, a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in constructing the subspace Hamiltonian (e.g., Remainder = idesired - Pnodes). Once the K basis states are determined for construction of the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Iterative Contribution-Based Selection Protocol

[0098] Figure 4F is a flowchart of a method 400f for implementing an iterative contributionbased selection protocol, according to embodiments of the present disclosure. Stated differently, method 400f applies an iterative contribution-based criterion for selection of sampled basis states to include from the Hilbert space of the full Hamiltonian, to construct thesubspace Hamiltonian, in which the iterative contribution-based criterion is satisfied based on applying a symmetry criterion to the set of N sampled basis states to obtain a set of S remaining basis states; sorting the set of S basis states bi based on their probabilities |<z2; making a trial selection of K basis states bi with the largest probabilities |<z2; constructing a subspace Hamiltonian HKusing the selected K basis states and solving for the eigensolution of HK,' selecting a subset of M basis states from the trial selection of K basis states based on the amplitudes of each basis state bi in the eigensolution of HKassigning a significance score to each basis state bi from S not included in the trial selection, where the significance score of each basis state is defined as sig(bt) = \eig(HM) — eig(HM+b^\; selecting the R basis states not included in the trial selection with the largest significance scores; and combining the selected sets of basis states Aland R.Block 41 Of- Sampling N Basis States from the Quantum Device

[0099] At block 41 Of, the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state bi having a probability |<z2of being sampled.Block 420f- Removing Basis States According to Symmetry Criterion and Sorting Basis States

[0100] At block 420f, the classical computing system removes the basis states from consideration for inclusion in the construction of the subspace Hamiltonian that do not satisfy a symmetry criterion, wherein the symmetry criterion comprises removing basis states that do not have bitstrings that contain a number of l’s that match the number of electrons in the chemical system under analysis, or comprises removing basis states that do not have bitstrings that have a number of l’s in positions corresponding to a given type of orbital equaling a number of electrons in the given type of orbital of a chemical system represented by the full Hamiltonian. As used herein the number of basis states that satisfy the symmetry criterion is referred to as S. The classical computer then sorts the basis states bi in S according to the probability |<z2for each bi.Block 430f- Trial Selection of K Basis States

[0101] At block 430f, the classical computing system makes a trial selection of the K basis states from the S remaining basis states with the highest probabilities |«;|2.Block 440f- Calculating Eigensolution of Trial Subspace Hamiltonian

[0102] At block 440f, the classical computing system constructs a subspace Hamiltonian HKwith the trial selection of K basis states from S, and solves the eigensolution of the subspace Hamiltonian.Block 450f- Selection of M Basis States from Trial Selection of K Basis States

[0103] At block 450f, the classical computing system selects a subset M of basis states bi from the trial selection of K basis states (where M < K) based on the amplitudes of each basis state bi in the eigensolution. Each amplitudeindicates how much basis state bi contributes to the eigensolution. The classical computing system then constructs a subspace Hamiltonian HMwith the AT selected basis states.Block 460f- Construction of Significance Scores for Basis States not in Trial Selection

[0104] At block 460f, the classical computer assigns a significance score to each basis state bi in S not included in the trial selection of K basis states at block 430f. The significance score of each basis state bi gives an indication of how important bi is to the eigensolution of subspace Hamiltonian HK. In various embodiments this significance score, herein referred to as sig(bi), is calculated as= \eig(HM) — eig(HM+bj\. This score compares the eigensolution of the subspace Hamiltonian HMwith the eigensolution of the subspace Hamiltonian constructed with basis states M+ bi, which indicates the effect of including bi in the selection of basis states.Block 465f- Selection of R Basis States using Significance Scores

[0105] At block 465f, the classical computing system selects from the set of S basis states, the set of R basis states with the R highest significance scores. This selection represents the set of R basis states bi with the highest observed impact on improving the eigensolution of the subspace Hamiltonian HM.Block 470f- Combining M Trial Basis States with R Basis States

[0106] At block 470f, the classical computing system combines the set of AT basis states bi with largest amplitudesin the eigensolution, with the set of R basis states with largest significance scores sig(bf. As used herein the number of basis states included in the set M+R is referred to as EBS. Accordingly, the classical computing system produces a set of (up to) K basis states that have been evaluated as being significant to the generation of a representative eigensolution for a chemical system.Block 480f- Supplemental Selection / Removal

[0107] At block 480f, up to K basis states for selection are chosen from the EBS basis states identified as being an electron preserving selection. When EBS=A (or when EBS=Aremainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when EBS > K, a supplemental selection process may be performed to identify a subset of the EBS basis states to remove from the selection to reduce the K basis states for use in constructing the subspace Hamiltonian. Similarly, when EBS < K , a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in constructing the subspace Hamiltonian (e.g., Remainder = Aiesn-ed - EBS). Once the K basis states are determined for construction of the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Alpha / Beta Swapping Supplementation Protocol

[0108] Figure 4G is a flowchart of a method 400g for implementing an alpha / beta swapping supplementation protocol, according to embodiments of the present disclosure. In method 400g, the basis states selected by the application of a selection protocol, or a sequence of selection protocols, are supplemented using additional basis states from the Symmetry Space. These additional states are constructed by separating the selected basis states into the alpha-and beta-configurations thereof, and then performing a permutation in which the alpha- and beta-configurations are swapped, thereby creating a new basis state. If a basis state with a certain alpha- and beta-configuration is found to be important to the eigensolution, the basis state with the correspondingly swapped alpha- and beta-configuration has often been found to also be important for the eigensolution.Block 410g - Sampling N Basis States from the Quantum Device

[0109] At block 410g, the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state bj having a probability | a j |2of being sampled.Block 420g -Application of Selection Method(s)

[0110] At block 420g, the classical computing system applies either a single selection method or a sequence of selection methods to identify a subset Emit of the Abasis states sampled from the quantum computing system which satisfy the selection protocol(s). This selection of states forms the Core Space, to which the additional states selected by method 400g are appended.Block 430g Permute Alpha- and Beta-Configurations to Construct Additional Basis States

[0111] At block 430g, the classical computing system separates each basis state bi in Emit into its alpha- and beta-configurationas and [f respectively. The classical computing system then performs the permutation of swapping the alpha- and beta-configurations to obtain a new basis state b for each basis state bi in Emit. This forms the set ESUpp containing the basis states b -Block 440g Add the Constructed Basis States to the Selected Set of Basis States

[0112] At block 440g, the classical computing systems combines the basis state selections Einit and ESUpp to form a selection EBS of unique basis states. Through the supplementation of the Core Space Emit with ESuPP, the Core Space is expanded to include important basis states that were not obtained through the sampling by the quantum computing system.Block 450g - Perform (Optional) Supplemental Selection / Removal

[0113] At block 450g, up to K basis states for selection are chosen from the basis states identified by the initial selection and further supplementation by the alpha / beta swapping method. When EBS=A (or when EBS= remainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when EBS > K, a supplemental selection process may be performed to identify a subset of the EBS basis states to reduce the selection to K basis states for use in constructing the subspace Hamiltonian. Similarly, when EBS < K, a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in constructing the subspace Hamiltonian (e.g., ^remainder = Adesn-ed - EBS). Once the K basis states are determined for construction of the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Heuristic Supplementation Protocol

[0114] In method 400h, a heuristic is used to guide the selection of additional basis states from the Symmetry Space for inclusion in the Core Space. The heuristic used for each basis state is the hamming distance of the basis state with respect to the Hartree-Fock basis state. The Hartree-Fock basis state is well-known to have a significant contribution to the eigensolution and the importance of a basis state to the eigensolution has been observed to often be inversely proportional to the degree by which the basis state in question differs from the Hartree-Fock basis state.Block 41 Oh Sampling N Basis States from the Quantum Device

[0115] At block 41 Oh, the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state bj having a probability | a j |2of being sampled.Block 420h - Application of Selection Protocol(s)

[0116] At block 420h, the classical computing system applies either a single selection protocol or a sequence of selection protocols, to identify a subset Emit of the N basis states sampled from the quantum computing system which satisfy the selection protocol(s). This selection of states forms the Core Space to which the additional states selected by method 400h are appended.Block 430h - Select a Random Subset of Valid Basis States

[0117] At block 430h, the classical computing system selects a random subset Erand of basis states belonging to the Symmetry Space that are not contained in Einit. The selection Erand forms the domain of the probabilistic sampling in the following operations of method 400h.Block 440h - Calculate the Hamming Distance for Each Basis State

[0118] At block 440h, the classical computing system calculates the hamming distance hi with respect to the Hartree-Fock basis state for each basis state bi in Erand. The hamming distance hi for basis state bi is the number of positions at which bi differs from the Hartree-Fock basis state.Block 450h - Construct Probability Distribution for Sampling

[0119] At block 450h, the classical computing system assigns a sampling probability=for each basis state btin Erand, where w, = — . A number of basis states are then sampledfrom Erand to supplement the selection Einit, and the sampling probabilities p, are defined such that the probability of sampling a basis state bi is directly proportional to its similarity to the Hartree-Fock basis state.Block 460h - Sample Basis States from the Selected Subset

[0120] At block 460h, the classical computing system performs a sampling of basis states from Erand based on the probability distribution formed by the pt ’s, forming the set of sampled basis states ESUpp which supplement the Core Space Emit formed in block 420h.Block 47 Oh Add the Sampled Basis States to the Selected Set of Basis States

[0121] At block 470h, the classical computing systems combines the basis state selections Einit and ESUpp to form a selection EBS of unique basis states. Through the supplementation of the Core Space Emit with ESUpp, the Core Space is expanded to include important basis states that were not obtained through the sampling by the quantum computing system.Block 480h - Perform (Optional) Supplemental Selection

[0122] At block 480h, the K basis states for selection are chosen from the basis states identified by the initial selection and further supplementation by the heuristic supplementation method. When EBS=A (or when EBS= remainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when EBS > K, a supplemental selection process may be performed to identify a subset of the EBS basis states to reduce the selection to K basis states for use in constructing the subspace Hamiltonian. Similarly, when EBS < K, a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in constructing the subspace Hamiltonian (e.g., ^remainder = 'desn-ed - EBS). Once the K basis states are determined for construction of the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Fixed-K-Greatest Selection Protocol

[0123] Figure 41 is a flowchart of a method 400i for implementing a fixed-A-greatest selection protocol, with a fixed value for K in which the K basis states with the highest (or highest absolute) values from the A basis states sampled from the quantum device. In various embodiments, despite the advantages of being able to increase or decrease the size of the value used for K across iterations, having a known fixed value for K offers a simple solution when a known number of basis states are needed. For example, if after performing one or more of the selection protocols set forth herein in which resulted in fewer than Adesired basis states being selected, the system determines that a supplemental selection should be performed, and may perform method 400i on a fixed value of K (e.g., Kfixed, where Aed = Xdesired - Selected) to meet the number of basis states to add to the Selected basis states from one or more previous iterations to meet the number of basis states to select for Aiesn-ed. Similarly, if after performing one or more of the selection protocols set forth herein in which resulted in more than Aiesn-ed basis states being selected, the system determines that a supplemental removal should be performed, and may perform method 400i using a fixed value of K to retain only desired basis states fromthe initial selection.Block 410i Sampling N Basis States from the Quantum Device

[0124] At block 41 Oi, the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state bj having a probability | |2of being sampled.Block 420i - Sorting Basis States based on Probabilities

[0125] At block 420i, the classical computing system sorts the basis states bj sampled per block 410i based on the probability values |a2thereof, from highest to lowest. In various embodiments, various sorting algorithms may be used (e.g., bubble sort, heap sort, merge sort, tree sort, insertion sort, shell sort, etc.).Block 430i - Select K highest-value Basis States

[0126] At block 430i, the classical computing system selects the K basis states sorted as having the K highest values.Conceptual Illustration of the Fixed-K-Greatest Selection Protocol

[0127] Figure 51 illustrates an example cutoff for selection of the K basis states from the sorted basis states according to method 400i. In the illustrated example, a cutoff 525 for K is fixed at four (e.g., K=4) so that the four basis states with the greatest samples probabilities are selected, and the other basis states are excluded / omitted. Several basis states 535a-j are illustrated as being sorted with highest to lowest probability with four basis states 535a-d on the included-side of the cutoff 525 and six basis states 535e-j on the excluded-side of the cutoff 525. As will be appreciated, the identities of the basis states 535a-j are given for convenience to the reader and correspond to the sorted probability values for those basis states, which may have no bearing on the position of that basis state’s matrix elements in the Hamiltonian matrix representation.Considerations for use of Fixed-K-Greatest Selection Protocol

[0128] As will be appreciated, although convenient for selecting a pre-defined numbed of basis states, the Fixed-K-Greatest Selection Protocol focuses on a single selection criterion; sampling probability. Accordingly, the third basis state 535c may be selected for inclusion over the fifth basis state 535e despite the fifth basis state 535e having greater effect on the accuracy of the calculations than the third basis state 535c according to one or more other selectioncriteria (e.g., those discussed in relation to methods 400a-f). Similarly, the Fixed-K-Greatest Selection Protocol may also result in basis states with equal values being split between being included or excluded sets (e.g., basis state 535d and basis state 535e) from the included set of basis states. Additionally, sorting algorithms are generally recognized to be computationally intensive and sorting a large number of basis states (e.g., 2n) to generate an ordered listing to which a cutoff 525 can be applied may be time and resource intensive.Threshold-Probability Selection Protocol

[0129] Figure 4J is a flowchart of a method 400j for implementing a threshold-probability selection protocol, wherein the basis states with the values above a given threshold are selected from the basis states sampled from the quantum device. In various embodiments, the number of basis states selected may be much greater than or much lower than Xdesired, and various thresholds may be examined until the number of selected basis states is within a desired range of desired (e.g., desired ±10%). Beneficially, the threshold-probability selection protocol, despite operating with an unknown value ^selected, is computationally simple to perform (e.g., avoiding the need to perform computationally complex sorting algorithms; thereby allowing operation on an unsorted list or pool of basis states), and allows for rapid repetition using various different values for the probability threshold. Accordingly, the threshold-probability selection protocol may be used as an initial selection protocol (with supplementation or omission determinations made by other selection protocols) or may be used as a supplementation or omission protocol for another selection protocol with low computational overhead.Block 410j - Sampling N Basis States from the Quantum Device

[0130] At block 41 Oj , the quantum computing system samples a trial wave function to obtain the set of N basis states from among the 2nbasis states, with each basis state bj having a probability | I2of being sampled.Block 420j - Sorting Basis States based on Probabilities

[0131] At block 420j, the classical computing system selects all those basis states sampled per block 41 Oj that have a probability value above a threshold value.Block 430j - Determine Whether Kseiected is Close to Koesired

[0132] At block 43 Oj, the classical computing system determines whether the number of basis states selected per block 420j (K selected) is close to the number of basis states desired for selection ^Desired). When Kseiected is within a predefined window of Kdesired, method 400j mayproceed to block 440j . When Kseiected is not within a predefined window of Kdesired, method 400j may return to block 420j to select all the basis states above a different threshold value, or may proceed to another selection protocol.Block 440j - Supplemental Selection / Removal

[0133] At block 440j, the Aiected basis states, once close to desired or after determining that a threshold selection protocol is inapt for reaching fdesn-ed in calculating the eigensolution, are supplemented to reach ^desired. When Gelected= desired (or when ^selected=Xremainder from an unfilled portion of K from an earlier selection process), the selection process may conclude. However, when Greeted > A’desired, a supplemental selection process may be performed to identify a subset of the Selected basis states to remove from the initial selection W, elected of basis states for use in constructing the subspace Hamiltonian to reduce Selected to Adesired. Similarly, when ^selected < desired, a supplemental selection process may be performed to identify additional basis states to include in the K basis states for use in constructing the subspace Hamiltonian. Once the K basis states are determined for construction of the subspace Hamiltonian (using one or more selection protocols), the system may then calculate and output the eigensolution using those K basis states.Conceptual Illustration of the Threshold-Probability Selection Protocol

[0134] Figure 5 J illustrates examples of thresholds (T) for selection of the K basis states from the unsorted basis states according to method 400j. In the illustrated example, two thresholds 545a, 545b are shown where the first threshold 545a is greater than the second threshold 545b (e.g., x > y). For ease of understanding and comparison to Figure 51, the basis states 535a-j (and probabilities thereof) are given identically between Figure 51 and Figure 5J. As is illustrated in Figure 5 J, when T=x for the first threshold 545a, three basis states 535a-c are selected, and when T=y for the second threshold 545b, five basis states 535a-e are selected. As will be appreciated, the basis states 535a-j are in a different order in Figure 5 J than in Figure 51, thereby permitting the computer systems to omit performing a sorting algorithm to select the basis states.Threshold selection

[0135] In various embodiments, the value for the threshold may be based on a user-defined value or may be based on a derived value from the sampled basis states, such as a percentile probability value. In some embodiments, basis states with outlier values or those identified for inclusion a priori (e.g., from an earlier selection process) may be excluded from thecalculations when deriving a threshold. For example, x in Figure 5J may correspond to the 60thpercentile so that the top 40% of the basis states by probability value are selected, while y in Figure 5 J may correspond to the 50thpercentile so that the half of basis states with the highest probability values are selected. As will be appreciated, the percentile-based threshold may not always select a number of basis states that equal the percentage of the total selected basis states. Continuing the example where x is set to the 60thpercentile, only 30 percent of the total basis states in Figure 5J (e.g., three of the ten illustrated basis states 535a-j) have been selected due to there not being four (e.g., 40% of the ten illustrated basis states 535a-j ) with probabilities of x, despite x mathematically being equal to the 60thpercentile.Additional Selection Processes

[0136] As discussed in relation to the electron preserving selection processes (e.g., a symmetry criterion), the overlap partition selection processes (e.g., an overlap criterion), the threshold-based selection processes, etc., various supplemental selections may be made when one selection process results in more or fewer basis state than K. In various embodiments, an operator may re-run the selection process using a different selection criterion, or supplement a first selection criterion with a second selection criterion. In addition to or alternatively to the electron preserving and the overlap partition selection processes, an operator may select basis states for inclusion (or exclusion) in the K selected basis states randomly, via manual selection, or a threshold value for the amplitudes of the basis states (e.g., a screening criterion for values of the amplitudes that are above a given threshold). Additionally or alternatively, the operator may use values selected from a previous iteration of the selection process (or different samplings of values from the trial wave function or different preparations of a trial wave function).Benefits provided by Selection

[0137] By not just selecting how many basis states to include from those sampled from the Krylov subspace in the construction of the subspace Hamiltonian (e.g., K), but by also selecting which basis states to include from those sampled from the Krylov subspace in the construction of the subspace Hamiltonian via one or more selection protocols, an operator can alter how representative the subspace Hamiltonian is of the full Hamiltonian. Accordingly, by being able to alter both the size of the subspace Hamiltonian (via adjusting K) and how the basis states with which to construct the subspace Hamiltonian are selected (via choosing different selection protocols), an operator can readily explore and test out different settings to allow for moreefficient or more accurate calculation of an eigensolution. Each of these selections, either for K or the selection protocol, can be performed independently of the other through an iterative process, such as is described in relation to method 600 described with respect to Figure 6.Flowchart Overview of Krylov Handover Procedure per Method 600

[0138] Figure 6 is a flowchart of an example method 600 for performing a Krylov Handover when determining the ground state value and ground state energy of a Hamiltonian, as may be used to represent a chemical system, according to embodiments of the present disclosure.Full Hamiltonian Generation

[0139] Method 600 begins at block 610, where a full Hamiltonian is created to represent the chemical system. A full Hamiltonian representing a chemical system is created using the classical computing system. The full Hamiltonian has a Hilbert space with 2nbasis states, which represent all possible states that the quantum system can be in. This is in contrast to the subspace Hamiltonian, which includes a subset of the 2nbasis states. Hamiltonians are used to evaluate the energy of the states the chemical system can occupy, and the subspace Hamiltonian offers a computationally simpler platform to evaluate that energy state, which the classical computing system can evaluate, which is in contrast to the full Hamiltonian, which can be too complex for a classical computing system to process in a reasonable timeframe, and therefore uses the quantum computing system to evaluate.Krylov Subspace State Generation

[0140] Method 600 then proceeds through one or more iterations of blocks 620a-Sk to prepare a first through SkthKrylov state in a Krylov subspace. Each of the different Krylov states use different setup parameters based on the full Hamiltonian, which are sampled to yield various basis states. The sampling operation is performed once per iteration of method 600, but may include sampling multiple times from each Krylov state, which yields a corresponding number Sk of sets of sampled basis states with one set per Krylov state. Each set of the Sk sets holds one or more basis states sampled from the associated Krylov state during the iteration’s sampling operation.Selection of Sampled Basis States to Include in Analysis Selection

[0141] At block 630, the quantum device samples a set of basis states from each Krylov state in the Krylov subspace, and at block 640, the system processes the sets of sampled basis states to develop an analysis selection of K basis states. From the Sk sets of sampled basis states, Kbasis states are selected according to various advanced heuristics and selection protocols such as those discussed in relation to Figures 4A-4J. These heuristics can reduce the number of basis states to K, or supplement the number of basis states up to K.Subspace Hamiltonian Generation

[0142] At block 650, from the K selected basis states, the classical computing system constructs a subspace Hamiltonian HK, and classically diagonalizes the subspace Hamiltonian HK at block 660 to obtain a target eigenvalue c and eigenstate CK. When the value for the target eigenvalue c converges, the method 600 may end, but otherwise method 600 proceeds from block 670 to block 680, where the initial state used on the quantum circuit is updated using the latest value for the target eigenstate CK. The classical computer then updates the initial state per block 680 such that |yo) =|\| / K , and method 600 returns to blocks 620a-Sk to prepare new sampling states based on the new initial state for the next iteration of blocks 620-670. The Krylov Handover procedure obeys the variational principle, so the eigenvalue will not be lower than the true eigenvalue of the target eigenstate of the full Hamiltonian.Initial States of Quantum Circuit Differing Across Iterations

[0143] In various embodiments, for the initial iteration of method 600 the classical computing system assigns an initial state for the chemical system for the quantum computing system to begin iteratively analyzing. The initial state represents an initial assignment of the amplitudes for the full set of 2nbasis states to begin calculations from. In various embodiments, the classical computing system may use various state preparation protocols for assigning the initial state for the chemical system. This initial state may include at least one of: a state prepared via a 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 (e.g., a Density Functional Theory (DFT), Configuration Interaction (CI), Coupled Cluster (CC), Moller-Plesset Perturbation theory (MPn), etc.), a state prepared via a tensor-network initial state preparation protocol (e.g., optimizing a tensor network and mapping the resulting state to a quantum circuit to obtain a quantum state), a sparse initial state from an eigensolution from a final iteration of a previous set of iterative analyses by the hybrid computing system (e.g., a final iteration of a previous plurality of iterations for use in a subsequent plurality of iterations), a uniformly distributed state, and a randomly distributed state. The present disclosure contemplates that different protocols may be used when performing subsequent iterations by the quantum computing system on the same chemical system, which may include theeigensolution from the just-completed iteration.Energy of Chemical Systems with Noiseless Model

[0144] Figures 7A-7C illustrate analysis of chemical systems of water (H2O), methane (CH4), and hydrogen peroxide (H2O2), respectively, according to a Krylov handover process as described herein with the quantum states developed using a quantum computer simulator with 12 qubits, 8 electrons, and 10,000 shots for water, 16 qubits, 8 electrons, and 100,000 shots formethane, and 18 qubits, 12 electrons, and 100,000 shots for hydrogen peroxide. These analyses used the STO-3G basis set and a noiseless classical simulation model.Classical Computing Device

[0145] Figure 8 illustrates a classical computing device 800, according to embodiments of the present disclosure. The classical computing device 800 may include a processor 810, a memory 820, and a communication interface 830.Processor

[0146] The processor 810 may be any processing unit capable of performing the operations and procedures described in the present disclosure according to directions or instructions. In various embodiments, the processor 810 can represent a single processor, multiple processors, a processor with multiple cores, and combinations thereof.Memory

[0147] The memory 820 is an apparatus that may be either volatile or non-volatile memory and may include RAM, flash, cache, disk drives, and other computer readable memory storage devices. Although shown as a single entity, the memory 820 may be divided into different memory storage elements such as RAM and one or more hard disk drives. As used herein, the memory 820 is an example of a device that includes computer-readable storage media, and is not to be interpreted as transmission media or signals per se.Inclusion of Executable Instructions in Memory

[0148] As shown, the memory 820 includes various instructions that are executable by the processor 810 to provide an operating system 822 to manage various features of the classical computing device 800 and one or more programs 824 to provide various functionalities to users of the classical computing device 800, which include one or more of the features and functionalities described in the present disclosure.Programming choices do not require undue experimentation

[0149] One of ordinary skill in the relevant art will recognize that different approaches can be taken in selecting or designing a program 824 to perform the operations described herein, including choice of programming language, the operating system 822 used by the classical computing device 800, and the architecture of the processor 810 and memory 820. Accordingly, the person of ordinary skill in the relevant art will be able to select or design an appropriate program 824 based on the details provided in the present disclosure.Peripheral Devices

[0150] The communication interface 830 facilitates communications between the classical computing device 800 and other devices, which may also be computing devices as described in relation to Figure 8. In various embodiments, the communication interface 830 includes antennas for wireless communications and various wired communication ports. The classical computing device 800 may also include or be in communication, via the communication interface 830, one or more input devices (e.g., a keyboard, mouse, pen, touch input device, etc.) and one or more output devices (e.g., a display, speakers, a printer, etc.).Networking Computing Devices together

[0151] Although not explicitly shown in Figure 8, it should be recognized that the classical computing device 800 may be connected to one or more public and / or private networks via appropriate network connections via the communication interface 830. It will also be recognized that software instructions may also be loaded into a non-transitory computer readable medium, such as the memory 820, from an appropriate storage medium or via wired or wireless means.Restatement of classical computing device

[0152] Accordingly, the classical computing device 800 is an example of a system that includes a processor 810 and a memory 820 that includes instructions that (when executed by the processor 810) perform various embodiments of the present disclosure. Similarly, the memory 820 is an apparatus that includes instructions that, when executed by a processor 810, perform various embodiments of the present disclosure.Quantum Computing Device

[0153] Figure 9 illustrates a quantum computing device 900, according to embodiments of the present disclosure. The quantum computing device 900 may include at least acommunications interface 910, a quantum programming interface 920, a quantum state preparation circuitry 930, a quantum computing circuitry 940, and a quantum control and measurement circuitry 950.Communications Interface

[0154] The communications interface 910 facilitates communications between the quantum computer 900 and other devices, which may include a classical computer (e.g., such as the classical computing device 800 discussed in relation to Figure 8 when providing a client accessing the quantum computing device 900 for a quantum application). In various embodiments, the communications interface 910 includes antennas for wireless communications and various wired communication ports. The classical computer pre-processes and sends input data (e.g., parameters, configurations, etc.) to the quantum computing device 900, retrieves and post-processes output data from the quantum computing device 900, and finally interprets and presents results to end users.Quantum Computers

[0155] Quantum computers are designed to solve problems that classical computers cannot solve efficiently by harnessing the laws of quantum mechanics. Although still in the early stages, quantum computers demonstrate great promises in boundless potential applications, including cryptography, artificial intelligence, drug discovery, material science, scientific research and simulations, and countless other fields. Quantum computers and classical computers differ fundamentally in underlying principles, computational models, and capabilities. Unlike classical computers, which process information in bits that can only represent one of two binary states at a time with a deterministic logic, quantum computers process information in a plurality of quantum bits (qubits) that can represent a coherent superposition of both binary states at the same time. In addition, two or more qubits may be entangled with each other, leading to highly correlated quantum states to enable massive quantum parallelism so that a vast number of operations can be performed simultaneously. Quantum computers can further use quantum interference to amplify the probability of obtaining correct results and suppress the probability of obtaining wrong results. Compared to classical computers, quantum computers have the potential to achieve exponentially faster calculation speeds, and address extremely complex problems with high accuracy that is beyond the capability of today’s most powerful (classical) supercomputers.Quantum Computer Hardware

[0156] The hardware used to provide the qubits may vary in different embodiments to match the application requirements for a given quantum computing device. In some embodiments, the qubits encode the quantum states via photons (polarization), coherent states of light, electrons (spin), a nucleus of an atom or chemical compound, a trapped ion, quantum dots, and the like, many of which are still being developed. The illustrated quantum computing device 900 is therefore provided as a general purpose representation of a quantum computing device 900 that may implement qubits and affect state changes therein via various hardware technologies.Context

[0157] Although not explicitly shown in Figure 9, the present disclosure contemplates that the quantum computing device 900 may be connected to one or more public and / or private networks via appropriate network connections via the communications interface 910. The present disclosure also contemplates that software instructions may also be loaded into a non-transitory computer readable medium, from an appropriate storage medium or via wired or wireless means.Quantum Programming Interface

[0158] The quantum programming interface 920 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 920 provides a representation of the underlying quantum hardware details, which enables clients to access the resources of the quantum computing device 900 regardless of the specific hardware implementations (e.g., client agnostic quantum computing). As illustrated in Figure 9, by using the quantum programming interface 920, at the beginning of quantum processing, quantum inputs are generated, which represent properly formatted data suitable for further quantum computing, and at the end of quantum processing, the quantum output data are converted to client readable format and sent to the client through the communications interface 910.Quantum State Preparation Circuitry

[0159] The quantum state preparation circuitry 930 further encodes the quantum inputs to generate quantum states, which represent the initial states of qubits, such as superposition, entanglement, probability interpretation, and continuous evolution. Quantum states encode the information and properties of the quantum system, govern the behavior of physical elements, and form a basis for quantum algorithms.Quantum Computing Circuitry

[0160] The quantum computing circuitry 940 implements quantum algorithms and performs quantum computations using quantum gates driven according to the quantum states. The quantum computing circuitry further includes a quantum memory 942, a quantum processing unit 944 and a quantum error detection and correction 946.Quantum Memory

[0161] The quantum memory 942 stores and preserves multiple quantum states in various superposition arrangements for a certain period of time. The quantum processing unit 944 is an integral part which works on the quantum computer principles to accomplish the task based on quantum mechanics. The quantum processing unit 944 also stores the state of computations in terms of quantum mechanical states and uses quantum buses to communicate amongst various other units of the quantum computer 900. The quantum error detection and correction 946 locates and corrects errors that exist during the operations of the quantum computing due to noise and decoherence.Quantum Control And Measurement Circuitry

[0162] As illustrated in Figure 9, the quantum control and measurement circuitry 950 controls and monitors the operations of the quantum state preparation circuitry 930 and the quantum computing circuitry 940 to assists in error detection and correction process. At the end of quantum computing, the quantum control and measurement circuitry 950 measures the results so quantum outputs are available to be converted to client readable formats using the quantum programming interface 920 and provided to client through the communications interface 910.Overall Operation

[0163] Accordingly, the quantum computer 900 is an example of a system that includes a quantum programming interface 920 and a set of quantum circuitries (930, 940, and 950) to accomplish a complex quantum computing task received from a client through a communication interface 910.Additional Understanding of Combinations of Embodiments

[0164] In addition to the embodiments described above, many examples of specific combinations are within the scope of the disclosure, some of which are discussed in Figures 10A-10S, and some of which are detailed below:Clause 1:

[0165] A method with improved computational system efficiency and accuracy in calculating Hamiltonian eigensolutions, comprising: preparing a plurality of Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system; sampling a plurality of basis states from each Krylov state of the plurality of Krylov states in the Krylov subspace to produce a plurality of sampled basis states; processing the plurality of sampled basis states to yield an analysis selection; constructing a subspace Hamiltonian from the analysis selection; computing, via an eigensolver, an eigensolution for the chemical system from the subspace Hamiltonian; and outputting the eigensolution for the chemical system.Clause 2:

[0166] The method of any of clause 1 and 3-9, further comprising: updating an initial state of a quantum circuit used to generate the plurality of Krylov states during a first iteration that resulted in the eigensolution from the initial state to the eigensolution; preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system; sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states; processing the second plurality of sampled basis states to yield a second analysis selection; constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; and outputting the second eigensolution for the chemical system.Clause 3:

[0167] The method of any of clauses 1-2 and 4-9, wherein the initial state is based on a Hartree-Fock state of the chemical system.Clause 4:

[0168] The method of any of clauses 1-3 and 5-9, wherein a number of members of the plurality of Krylov states is different than a number of members of the plurality of second Krylov states.Clause 5:

[0169] The method of any of clauses 1-4 and 6-9, wherein the plurality of sampled basis states are orthonormal to one another.Clause 6:

[0170] The method of any of clauses 1-5 and 7-9, wherein processing the plurality of sampled basis states includes: selecting K basis states according to a selection protocol to define the final selection for the chemical system, wherein the selection protocol identifies individual basis states from the plurality of basis states to include in the final selection using at least one selection criterion other than a fixed-K-greatest selection protocol in which the individual basis state would be selected to be among the K basis states based on a magnitude of a probability of the individual basis state being among the K-highest probabilities in an ansatz space of the chemical system.Clause 7:

[0171] The method of any of clauses 1-6 and 8-9, wherein the plurality of Krylov states are each prepared using a different power of the full Hamiltonian.Clause 8:

[0172] The method of any of clauses 1-7 and 8, wherein the plurality of Krylov states are each prepared using different real-time evolutions of the full Hamiltonian.Clause 9:

[0173] The method of any of clauses 1-8, wherein a time-step parameter for the real-time evolutions is a different time step for each Krylov state of the plurality of Krylov states in the Krylov subspace.Clause 10:

[0174] A method with improved computational system efficiency and accuracy in calculating Hamiltonian eigensolutions, comprising: preparing a plurality of first Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system according to an initial state of a quantum circuit, wherein each first Krylov is prepared using a different power of the full Hamiltonian than each other or are each prepared using different real-time evolutions of the full Hamiltonian than each other; sampling a first plurality of basis states from each first Krylov state of the plurality of first Krylov states in the Krylov subspace to produce a first plurality of sampled basis states that are orthonormal to one another; processing the first plurality of sampled basis states to yield a first analysis selection of K basis states; constructing a first subspace Hamiltonian from the first analysis selection; computing, via an eigensolver, a first eigensolution for the chemical system from the first subspace Hamiltonian; outputting afirst eigensolution for the chemical system; updating the initial state of the quantum circuit based on the first eigensolution; preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system, wherein each second Krylov is prepared using a different power of the full Hamiltonian than each other or are each prepared using different real-time evolutions of the full Hamiltonian than each other; sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states that are orthonormal to one another; processing the second plurality of sampled basis states to yield a second analysis selection of K basis states; constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; and outputting the second eigensolution for the chemical system.Clause 11:

[0175] A method with improved computational system efficiency and accuracy in calculating Hamiltonian eigensolutions, comprising: preparing a plurality of Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system; sampling a plurality of basis states from each Krylov state of the plurality of Krylov states in the Krylov subspace to produce a plurality of sampled basis states; processing the plurality of sampled basis states to yield an analysis selection; constructing a subspace Hamiltonian from the analysis selection; computing, via an eigensolver, an eigensolution for the chemical system from the subspace Hamiltonian; and outputting the eigensolution for the chemical system.Clause 12:

[0176] The method of clause 11, further comprising: updating an initial state of a quantum circuit used to generate the plurality of Krylov states during a first iteration that resulted in the eigensolution from the initial state to the eigensolution; preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system; sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states; processing the second plurality of sampled basis states to yield a second analysis selection; constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; and outputting the second eigensolution for the chemical system.Clause 13:

[0177] The method of clause 12, wherein the initial state is based on a Hartree-Fock state of the chemical system.Clause 14:

[0178] The method of clause 12, wherein a number of members of the plurality of Krylov states is different than a number of members of the plurality of second Krylov states.Clause 15:

[0179] The method of clause 11, wherein the plurality of sampled basis states are orthonormal to one another.Clause 16:

[0180] The method of clause 11, wherein processing the plurality of sampled basis states includes: selecting K basis states according to a selection protocol to define the final selection for the chemical system, wherein the selection protocol identifies individual basis states from the plurality of basis states to include in the final selection using at least one selection criterion other than a fixed-K-greatest selection protocol in which the individual basis state would be selected to be among the K basis states based on a magnitude of a probability of the individual basis state being among the K-highest probabilities in an ansatz space of the chemical system.Clause 17:

[0181] The method of clause 11, wherein the plurality of Krylov states are each prepared using a different power of the full Hamiltonian.Clause 18:

[0182] The method of clause 11, wherein the plurality of Krylov states are each prepared using different real-time evolutions of the full Hamiltonian.Clause 19:

[0183] The method of clause 18, wherein a time-step parameter for the real-time evolutions is a different time step for each Krylov state of the plurality of Krylov states in the Krylov subspace.Clause 20:

[0184] The method of clause 11, wherein an initial state the full Hamiltonian that representsthe chemical system is prepared for initially representing the Krylov space, wherein the initial state comprises at least one of: a previously calculated eigensolution generated from a previous Krylov space prepared for the full Hamiltonian; a state prepared via a 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 uniformly distributed state; and a randomly distributed state.Clause 21:

[0185] The method of clause 11, further comprising: modifying the full Hamiltonian according to a rule to remove terms of a Pauli representation of the full Hamiltonian.Clause 22:

[0186] The method of clause 21, wherein the rule comprises removing terms of the Pauli representation of the full Hamiltonian having coefficients below a retention threshold.Clause 23:

[0187] The method of clause 21, further comprising, applying an initial state modification scheme selected from the group consisting of: updating an initial state every iteration of calculating the eigensolution; updating the initial state for a subset of Krylov states in the Krylov subspace every iteration of calculating the eigensolution; and using a different initial state for each Krylov state in the Krylov subspace in a given iteration of calculating the eigensolution.Clause 24:

[0188] The method of clause 21, further comprising, controlling values selected for defining a time term t that specified a time step for evolution of the eigensolution, selected from the group consisting of: using a different time step for each Krylov state in the Krylov subspace; using a same time step for each Krylov state in the Krylov subspace; selecting an a priori time step according to an a priori knowledge about the chemical system; and selected an optimized time step based on previous time step over several iterations of calculating the eigensolution.Clause 25:

[0189] A method with improved computational system efficiency and accuracy in calculating Hamiltonian eigensolutions, comprising: preparing a plurality of first Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system according to an initial state of a quantum circuit, wherein each first Krylov is prepared using a different powerof the full Hamiltonian than each other or are each prepared using different real-time evolutions of the full Hamiltonian than each other; sampling a first plurality of basis states from each first Krylov state of the plurality of first Krylov states in the Krylov subspace to produce a first plurality of sampled basis states that are orthonormal to one another; processing the first plurality of sampled basis states to yield a first analysis selection of K basis states; constructing a first subspace Hamiltonian from the first analysis selection; computing, via an eigensolver, a first eigensolution for the chemical system from the first subspace Hamiltonian; outputting a first eigensolution for the chemical system; updating the initial state of the quantum circuit based on the first eigensolution; preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system, wherein each second Krylov is prepared using a different power of the full Hamiltonian than each other or are each prepared using different real-time evolutions of the full Hamiltonian than each other; sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states that are orthonormal to one another; processing the second plurality of sampled basis states to yield a second analysis selection of K basis states; constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; and outputting the second eigensolution for the chemical system.Clause 26:

[0190] A method, comprising: preparing a plurality of Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system, wherein each Krylov state of the plurality of Krylov states represents a quantum state of the chemical system prepared using a quantum circuit that implements a different implementation of a function of the Hamiltonian for each Krylov state of the plurality of Krylov states, applied to a first state of the quantum circuit; sampling a plurality of basis states from each Krylov state of the plurality of Krylov states in the Krylov subspace to produce a plurality of sampled basis states; processing the plurality of sampled basis states to yield an analysis selection; constructing a subspace Hamiltonian from the analysis selection; computing, via an eigensolver, an eigensolution for the chemical system from the subspace Hamiltonian; and outputting the eigensolution for the chemical system.Clause 27:

[0191] The method of clause 26, further comprising: updating the first state of the quantumcircuit used to generate the plurality of Krylov states during a first iteration that resulted in the eigensolution from the first state to a second state; preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system according the second state of the quantum circuit; sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states; processing the second plurality of sampled basis states to yield a second analysis selection; constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; and outputting the second eigensolution for the chemical system.Clause 28:

[0192] The method of clause 27, wherein the first state is based on a Hartree-Fock state of the chemical system.Clause 29:

[0193] The method of clause 27, wherein a number of members of the plurality of Krylov states is different than a number of members of the plurality of second Krylov states.Clause 30:

[0194] The method of clause 26, wherein the plurality of sampled basis states are orthonormal to one another.Clause 31:

[0195] The method of clause 26, wherein processing the plurality of sampled basis states includes: selecting K basis states according to a selection protocol to define the final selection for the chemical system, wherein the selection protocol identifies individual basis states from the plurality of basis states to include in the final selection using at least one selection criterion other than a fixed-K-greatest selection protocol in which the individual basis state would be selected to be among the K basis states based on a magnitude of a probability of the individual basis state being among the K-highest probabilities in an ansatz space of the chemical system.Clause 32:

[0196] The method of clause 26, wherein each Krylov state of the plurality of Krylov states is prepared using a different power of the full Hamiltonian than other Krylov states of the plurality of Krylov states.Clause 33:

[0197] The method of clause 26, wherein each Krylov state of the plurality of Krylov states is prepared using different real-time evolutions of the full Hamiltonian than other Krylov states of the plurality of Krylov states.Clause 34:

[0198] The method of clause 33, wherein a time-step parameter for the real-time evolutions is a different time step for each Krylov state of the plurality of Krylov states in the Krylov subspace.Clause35:

[0199] The method of clause 26, further comprising: simulating the chemical system in in a user-specified state according to the eigensolution and relative to a target chemical system; and determining a most stable configuration of the chemical system with respect to the target chemical system.Clause 36:

[0200] The method of clause 26, wherein a first one of the chemical system and the target chemical system include at least a portion of a biological molecule and a second one of the chemical system and the target chemical system include at least a portion of a pharmacological compound.Clause 37:

[0201] The method of clause 26, wherein an initial state the full Hamiltonian that represents the chemical system is used for preparing the Krylov space, wherein the initial state comprises at least one of: a previously calculated eigensolution generated from a previous Krylov space prepared for the full Hamiltonian; a state prepared via a 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 uniformly distributed state; and a randomly distributed state.Clause 38:

[0202] The method of clause 26, further comprising: modifying the full Hamiltonian according to a rule to remove terms of a Pauli representation of the full Hamiltonian.Clause 39:

[0203] The method of clause 38, wherein the rule comprises removing terms of a Pauli representation of the full Hamiltonian having coefficients below a retention threshold.Clause 40:

[0204] The method of clause 26, further comprising, applying an initial state modification scheme selected from the group consisting of: updating an initial state every iteration of calculating the eigensolution; updating the initial state for a subset of Krylov states in the Krylov subspace every iteration of calculating the eigensolution; and using a different initial state for each Krylov state in the Krylov subspace in a given iteration of calculating the eigensolution.Clause 41:

[0205] The method of clause 26, further comprising, controlling values selected for defining a time term t that specified a time step for evolution of the eigensolution, selected from the group consisting of: using a different time step for each Krylov state in the Krylov subspace; using a same time step for each Krylov state in the Krylov subspace; selecting an a priori time step according to an a priori knowledge about the chemical system; and selected an optimized time step based on previous time step over several iterations of calculating the eigensolution.Clause 42:

[0206] A system, comprising: a processor; and a memory, including instructions that, when executed by the processor, perform operations of the method of any of clauses 26-41.Clause 43:

[0207] A non-transitory memory storage apparatus, including instructions that, when executed by a processor, perform operations of the method of any of clauses 26-41.Clause 44:

[0208] A method, comprising: preparing a plurality of first Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system according to a first state of a quantum circuit, wherein each first Krylov state of the plurality of first Krylov states is prepared using a different power of the full Hamiltonian than each other or are each prepared using different real-time evolutions of the full Hamiltonian than each other; sampling a first plurality of basis states from each first Krylov state of the plurality of first Krylov states in the Krylov subspace to produce a first plurality of sampled basis states that are orthonormal to one another; processing the first plurality of sampled basis states to yield a first analysis selection of KIbasis states; constructing a first subspace Hamiltonian from the first analysis selection; computing, via an eigensolver, a first eigensolution for the chemical system from the first subspace Hamiltonian; outputting a first eigensolution for the chemical system; updating the first state of the quantum circuit based on the first eigensolution to a second state; preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system according to the second state of the quantum circuit; sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states that are orthonormal to one another; processing the second plurality of sampled basis states to yield a second analysis selection of K2 basis states; constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; and outputting the second eigensolution for the chemical system.Clause 45:

[0209] The method of clause 44, wherein each second Krylov state of the plurality of second Krylov states is prepared using a different power of the full Hamiltonian than each other second Krylov state of the plurality of second Krylov states.Clause 46:

[0210] The method of clause 44, wherein each second Krylov state of the plurality of second Krylov states is each prepared using different real-time evolutions of the full Hamiltonian than each other second Krylov state of the plurality of second Krylov states.Clause 47:

[0211] The method of clause 44, further comprising: simulating the chemical system in in a user-specified state according to the eigensolution and relative to a target chemical system; and determining a most stable configuration of the chemical system with respect to the target chemical system.Clause 48:

[0212] The method of clause 47, wherein a first one of the chemical system and the target chemical system include at least a portion of a biological molecule and a second one of the chemical system and the target chemical system include at least a portion of a pharmacological compound.Clause 49:

[0213] The method of clause 44, wherein an initial state the full Hamiltonian that represents the chemical system is used for preparing the Krylov space, wherein the initial state comprises at least one of: a previously calculated eigensolution generated from a previous Krylov space prepared for the full Hamiltonian; a state prepared via a 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 uniformly distributed state; and a randomly distributed state.Clause 50:

[0214] The method of clause 44, further comprising: modifying the full Hamiltonian according to a rule to remove terms of a Pauli representation of the full Hamiltonian.Clause 51:

[0215] The method of clause 50, wherein the rule comprises removing terms of a Pauli representation of the full Hamiltonian having coefficients below a retention threshold.Clause 52:

[0216] The method of clause 44, further comprising, applying an initial state modification scheme selected from the group consisting of: updating an initial state every iteration of calculating the eigensolution; updating the initial state for a subset of Krylov states in the Krylov subspace every iteration of calculating the eigensolution; and using a different initial state for each Krylov state in the Krylov subspace in a given iteration of calculating the eigensolution.Clause 53:

[0217] The method of clause 44, further comprising, controlling values selected for defining a time term t that specified a time step for evolution of the eigensolution, selected from the group consisting of: using a different time step for each Krylov state in the Krylov subspace; using a same time step for each Krylov state in the Krylov subspace; selecting an a priori time step according to an a priori knowledge about the chemical system; and selected an optimized time step based on previous time step over several iterations of calculating the eigensolution.Clause 54:

[0218] A system, comprising: a processor; and a memory, including instructions that, when executed by the processor, perform operations of the method of any of clauses 44-53.Clause 55:

[0219] A non-transitory memory storage apparatus, including instructions that, when executed by a processor, perform operations of the method of any of clauses 44-53.Description of Terminology used in the Present Disclosure

[0220] Certain terms are used throughout the description and claims to refer to particular features or components. As one skilled in the art will appreciate, different persons may refer to the same feature or component by different names. This document does not intend to distinguish between components or features that differ in name but not function.Interpretation of “Optimize ”

[0221] As used herein, the term “optimize” and variations thereof, is used in a sense understood by data scientists to refer to actions taken for continual improvement of a system relative to a goal. An optimized value will be understood to represent “near-best” value for a given reward framework, which may oscillate around a local maximum / minimum or a global maximum / minimum for a “best” value or set of values, which may change as the goal changes or as input conditions change. Accordingly, an optimal solution for a first goal at a given time may be suboptimal for a second goal at that time or suboptimal for the first goal at a later time.Chemical Naming Conventions

[0222] As used herein, various chemical compounds are referred to by associated element abbreviations set by the International Union of Pure and Applied Chemistry (IUPAC), which one of ordinary skill in the relevant art will be familiar with. Similarly, various units of measure may be used herein, which are referred to by associated short forms as set by the International System of Units (SI), which one of ordinary skill in the relevant art will be familiar with.Interpretation of About, Approximately, and Substantially

[0223] As used herein, “about,” “approximately” and “substantially” are understood to refer to numbers in a range of the referenced number, for example the range of -10% to +10% of the referenced number, preferably -5% to +5% of the referenced number, more preferably -1% to +1% of the referenced number, most preferably -0.1% to +0.1% of the referenced number.Numerical Ranges

[0224] Furthermore, all numerical ranges herein should be understood to include all integers, whole numbers, or fractions, within the range. Moreover, these numerical ranges should be construed as providing support for a claim directed to any number or subset of numbers in thatrange. For example, a disclosure of from 1 to 10 should be construed as supporting a range of from 1 to 8, from 3 to 7, from 1 to 9, from 3.6 to 4.6, from 3.5 to 9.9, and so forth.Interpretation of “at least one of”

[0225] As used in the present disclosure, a phrase referring to “at least one of’ a list of items refers to any set of those items, including sets with a single member, and every potential combination thereof. For example, when referencing “at least one of A, B, or C” or “at least one of A, B, and C”, the phrase is intended to cover the sets of: A, B, C, A-B, B-C, A-C, and A-B-C, where the sets may include one or multiple instances of a given member (e.g., A-A, A-A-A, A-A-B, A-A-B-B-C-C-C, etc.) and any ordering thereof. For avoidance of doubt, the phrase “at least one of A, B, and C” shall not be interpreted to mean “at least one of A, at least one of B, and at least one of C”.Interpretation of “Determining”

[0226] As used in the present disclosure, the term “determining” encompasses a variety of actions that may include calculating, computing, processing, deriving, investigating, looking up (e.g., via a table, database, or other data structure), ascertaining, receiving (e.g., receiving information), accessing (e.g., accessing data in a memory), retrieving, resolving, selecting, choosing, establishing, and the like.Examples are illustrative and do not limit scope

[0227] Without further elaboration, it is believed that one skilled in the art can use the preceding description to use the claimed inventions to their fullest extent. The examples and aspects disclosed herein are to be construed as merely illustrative and not a limitation of the scope of the present disclosure in any way. It will be apparent to those having skill in the art that changes may be made to the details of the above-described examples without departing from the underlying principles discussed. In other words, various modifications and improvements of the examples specifically disclosed in the description above are within the scope of the appended claims. For instance, any suitable combination of features of the various examples described is contemplated.Claim construction

[0228] Within the claims, reference to an element in the singular is not intended to mean “one and only one” unless specifically stated as such, but rather as “one or more” or “at least one”. Unless specifically stated otherwise, the term “some” refers to one or more. No claimelement is to be construed under the provision of 35 U.S.C. § 112(f) unless the element is expressly recited using the phrase “means for” or “step for”. All structural and functional equivalents to the elements of the various embodiments described in the present disclosure that are known or come later to be known to those of ordinary skill in the relevant art are expressly incorporated herein by reference and are intended to be encompassed by the claims. Moreover, nothing disclosed in the present disclosure is intended to be dedicated to the public regardless of whether such disclosure is explicitly recited in the claims.

Claims

CLAIMSWe claim:

1. A method with improved computational system efficiency and accuracy in calculating Hamiltonian eigensolutions, comprising:preparing a plurality of Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system;sampling a plurality of basis states from each Krylov state of the plurality of Krylov states in the Krylov subspace to produce a plurality of sampled basis states;processing the plurality of sampled basis states to yield an analysis selection; constructing a subspace Hamiltonian from the analysis selection;computing, via an eigensolver, an eigensolution for the chemical system from the subspace Hamiltonian; andoutputting the eigensolution for the chemical system.

2. The method of claim 1, further comprising:updating an initial state of a quantum circuit used to generate the plurality of Krylov states during a first iteration that resulted in the eigensolution from the initial state to the eigensolution;preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system;sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states;processing the second plurality of sampled basis states to yield a second analysis selection;constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; andoutputting the second eigensolution for the chemical system.

3. The method of claim 2, wherein the initial state is based on a Hartree-Fock state of the chemical system.

4. The method of claim 2, wherein a number of members of the plurality of Krylov states is different than a number of members of the plurality of second Krylov states.

5. The method of claim 1, wherein the plurality of sampled basis states are orthonormal to one another.

6. The method of claim 1, wherein processing the plurality of sampled basis states includes:selecting T basis states according to a selection protocol to define a final selection for the chemical system,wherein the selection protocol identifies individual basis states from the plurality of basis states to include in the final selection using at least one selection criterion other than a fixed- T-greatest selection protocol in which the individual basis state would be selected to be among the K basis states based on a magnitude of a probability of the individual basis state being among the A"-highest probabilities in an ansatz space of the chemical system.

7. The method of claim 1, wherein the plurality of Krylov states are each prepared using a different power of the full Hamiltonian.

8. The method of claim 1, wherein the plurality of Krylov states are each prepared using different real-time evolutions of the full Hamiltonian.

9. The method of claim 8, wherein a time-step parameter for the real-time evolutions is a different time step for each Krylov state of the plurality of Krylov states in the Krylov subspace.

10. The method of claim 1, wherein an initial state the full Hamiltonian that represents the chemical system is prepared for initially representing the Krylov subspace, wherein the initial state comprises at least one of:a previously calculated eigensolution generated from a previous Krylov space prepared for the full Hamiltonian;a state prepared via a Hartree-Fock protocol;a zero state;a computational basis state of a 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 uniformly distributed state; anda randomly distributed state.

11. The method of claim 1, further comprising:modifying the full Hamiltonian according to a rule to remove terms of a Pauli representation of the full Hamiltonian.

12. The method of claim 11, wherein the rule comprises removing terms of the Pauli representation of the full Hamiltonian having coefficients below a retention threshold.

13. The method of claim 1, further comprising, applying an initial state modification scheme selected from the group consisting of:updating an initial state every iteration of calculating the eigensolution; updating the initial state for a subset of Krylov states in the Krylov subspace every iteration of calculating the eigensolution; andusing a different initial state for each Krylov state in the Krylov subspace in a given iteration of calculating the eigensolution.

14. The method of claim 1, further comprising, controlling values selected for defining a time term t that specified a time step for evolution of the eigensolution, selected from the group consisting of:using a different time step for each Krylov state in the Krylov subspace;using a same time step for each Krylov state in the Krylov subspace;selecting an a priori time step according to an a priori knowledge about the chemical system; andselected an optimized time step based on previous time step over several iterations of calculating the eigensolution.

15. A method with improved computational system efficiency and accuracy in calculating Hamiltonian eigensolutions, comprising:preparing a plurality of first Krylov states in a Krylov subspace of a full Hamiltonianthat represents a chemical system according to an initial state of a quantum circuit, wherein each first Krylov is prepared using a different power of the full Hamiltonian than each other or are each prepared using different real-time evolutions of the full Hamiltonian than each other;sampling a first plurality of basis states from each first Krylov state of the plurality of first Krylov states in the Krylov subspace to produce a first plurality of sampled basis states that are orthonormal to one another;processing the first plurality of sampled basis states to yield a first analysis selection of K basis states;constructing a first subspace Hamiltonian from the first analysis selection; computing, via an eigensolver, a first eigensolution for the chemical system from the first subspace Hamiltonian;outputting a first eigensolution for the chemical system;updating the initial state of the quantum circuit based on the first eigensolution; preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system, wherein each second Krylov is prepared using a different power of the full Hamiltonian than each other or are each prepared using different real-time evolutions of the full Hamiltonian than each other;sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states that are orthonormal to one another;processing the second plurality of sampled basis states to yield a second analysis selection of K basis states;constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; andoutputting the second eigensolution for the chemical system.

16. A method, comprising:preparing a plurality of Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system, wherein each Krylov state of the plurality of Krylov states represents a quantum state of the chemical system prepared using a quantum circuit that implements a different implementation of a function of the Hamiltonian for each Krylov state of the plurality of Krylov states, applied to a first state of the quantum circuit;sampling a plurality of basis states from each Krylov state of the plurality of Krylov states in the Krylov subspace to produce a plurality of sampled basis states;processing the plurality of sampled basis states to yield an analysis selection; constructing a subspace Hamiltonian from the analysis selection;computing, via an eigensolver, an eigensolution for the chemical system from the subspace Hamiltonian; andoutputting the eigensolution for the chemical system.

17. The method of claim 16, further comprising:updating the first state of the quantum circuit used to generate the plurality of Krylov states during a first iteration that resulted in the eigensolution from the first state to a second state;preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system according the second state of the quantum circuit;sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states;processing the second plurality of sampled basis states to yield a second analysis selection;constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; andoutputting the second eigensolution for the chemical system.

18. The method of claim 17, wherein the first state is based on a Hartree-Fock state of the chemical system.

19. The method of claim 17, wherein a number of members of the plurality of Krylov states is different than a number of members of the plurality of second Krylov states.

20. The method of claim 16, wherein the plurality of sampled basis states are orthonormal to one another.

21. The method of claim 16, wherein processing the plurality of sampled basis states includes:selecting K basis states according to a selection protocol to define a final selection for the chemical system,wherein the selection protocol identifies individual basis states from the plurality of basis states to include in the final selection using at least one selection criterion other than a fixed- T-greatest selection protocol in which the individual basis state would be selected to be among the K basis states based on a magnitude of a probability of the individual basis state being among the A"-highest probabilities in an ansatz space of the chemical system.

22. The method of claim 16, wherein each Krylov state of the plurality of Krylov states is prepared using a different power of the full Hamiltonian than other Krylov states of the plurality of Krylov states.

23. The method of claim 16, wherein each Krylov state of the plurality of Krylov states is prepared using different real-time evolutions of the full Hamiltonian than other Krylov states of the plurality of Krylov states.

24. The method of claim 23, wherein a time-step parameter for the real-time evolutions is a different time step for each Krylov state of the plurality of Krylov states in the Krylov subspace.

25. The method of claim 16, further comprising:simulating the chemical system in in a user-specified state according to the eigensolution and relative to a target chemical system; anddetermining a most stable configuration of the chemical system with respect to the target chemical system.

26. The method of claim 25, wherein a first one of the chemical system and the target chemical system include at least a portion of a biological molecule and a second one of the chemical system and the target chemical system include at least a portion of a pharmacological compound.

27. The method of claim 16, wherein an initial state the full Hamiltonian that representsthe chemical system is used for preparing the Krylov subspace, wherein the initial state comprises at least one of:a previously calculated eigensolution generated from a previous Krylov space prepared for the full Hamiltonian;a state prepared via a Hartree-Fock protocol;a zero state;a computational basis state of a 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 uniformly distributed state; anda randomly distributed state.

28. The method of claim 16, further comprising:modifying the full Hamiltonian according to a rule to remove terms of a Pauli representation of the full Hamiltonian.

29. The method of claim 28, wherein the rule comprises removing terms of a Pauli representation of the full Hamiltonian having coefficients below a retention threshold.

30. The method of claim 16, further comprising, applying an initial state modification scheme selected from the group consisting of:updating an initial state every iteration of calculating the eigensolution; updating the initial state for a subset of Krylov states in the Krylov subspace every iteration of calculating the eigensolution; andusing a different initial state for each Krylov state in the Krylov subspace in a given iteration of calculating the eigensolution.

31. The method of claim 16, further comprising, controlling values selected for defining a time term t that specified a time step for evolution of the eigensolution, selected from the group consisting of:using a different time step for each Krylov state in the Krylov subspace;using a same time step for each Krylov state in the Krylov subspace;selecting an a priori time step according to an a priori knowledge about the chemical system; andselected an optimized time step based on previous time step over several iterations ofcalculating the eigensolution.

32. A system, comprising:a processor; anda memory, including instructions that, when executed by the processor, perform operations of the method of claim 16.

33. A non-transitory memory storage apparatus, including instructions that, when executed by a processor, perform operations of the method of claim 16.

34. A method, comprising:preparing a plurality of first Krylov states in a Krylov subspace of a full Hamiltonian that represents a chemical system according to a first state of a quantum circuit, wherein each first Krylov state of the plurality of first Krylov states is prepared using a different power of the full Hamiltonian than each other or are each prepared using different real-time evolutions of the full Hamiltonian than each other;sampling a first plurality of basis states from each first Krylov state of the plurality of first Krylov states in the Krylov subspace to produce a first plurality of sampled basis states that are orthonormal to one another;processing the first plurality of sampled basis states to yield a first analysis selection of KI basis states;constructing a first subspace Hamiltonian from the first analysis selection; computing, via an eigensolver, a first eigensolution for the chemical system from the first subspace Hamiltonian;outputting a first eigensolution for the chemical system;updating the first state of the quantum circuit based on the first eigensolution to a second state;preparing a plurality of second Krylov states in the Krylov subspace of the full Hamiltonian that represents the chemical system according to the second state of the quantum circuit;sampling a second plurality of basis states from each second Krylov state of the plurality of second Krylov states in the Krylov subspace to produce a second plurality of sampled basis states that are orthonormal to one another;processing the second plurality of sampled basis states to yield a second analysisselection of K2 basis states;constructing a second subspace Hamiltonian from the second analysis selection; computing, via the eigensolver, a second eigensolution for the chemical system from the second subspace Hamiltonian; andoutputting the second eigensolution for the chemical system.

35. The method of claim 34, wherein each second Krylov state of the plurality of second Krylov states is prepared using a different power of the full Hamiltonian than each other second Krylov state of the plurality of second Krylov states.

36. The method of claim 34, wherein each second Krylov state of the plurality of second Krylov states is each prepared using different real-time evolutions of the full Hamiltonian than each other second Krylov state of the plurality of second Krylov states.

37. The method of claim 34, further comprising:simulating the chemical system in in a user-specified state according to the eigensolution and relative to a target chemical system; anddetermining a most stable configuration of the chemical system with respect to the target chemical system.

38. The method of claim 37, wherein a first one of the chemical system and the target chemical system include at least a portion of a biological molecule and a second one of the chemical system and the target chemical system include at least a portion of a pharmacological compound.

39. The method of claim 34, wherein an initial state the full Hamiltonian that represents the chemical system is used for preparing the Krylov subspace, wherein the initial state comprises at least one of:a previously calculated eigensolution generated from a previous Krylov space prepared for the full Hamiltonian;a state prepared via a Hartree-Fock protocol;a zero state;a computational basis state of a 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 uniformly distributed state; anda randomly distributed state.

40. The method of claim 34, further comprising:modifying the full Hamiltonian according to a rule to remove terms of a Pauli representation of the full Hamiltonian.

41. The method of claim 40, wherein the rule comprises removing terms of a Pauli representation of the full Hamiltonian having coefficients below a retention threshold.

42. The method of claim 34, further comprising, applying an initial state modification scheme selected from the group consisting of:updating an initial state every iteration of calculating the eigensolution; updating the initial state for a subset of Krylov states in the Krylov subspace every iteration of calculating the eigensolution; andusing a different initial state for each Krylov state in the Krylov subspace in a given iteration of calculating the eigensolution.

43. The method of claim 34, further comprising, controlling values selected for defining a time term t that specified a time step for evolution of the eigensolution, selected from the group consisting of:using a different time step for each Krylov state in the Krylov subspace;using a same time step for each Krylov state in the Krylov subspace;selecting an a priori time step according to an a priori knowledge about the chemical system; andselected an optimized time step based on previous time step over several iterations of calculating the eigensolution.

44. A system, comprising:a processor; anda memory, including instructions that, when executed by the processor, perform operations of the method of claim 34.

45. A non-transitory memory storage apparatus, including instructions that, when executed by a processor, perform operations of the method of claim 34.

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