Method and apparatus for fermionic local simulation of resource optimization on a quantum computer for quantum chemistry

Abstract

Aspects of the present disclosure describe a method comprising: predicting a first set of ansatz terms and a plurality of first amplitudes associated with the first set of ansatz terms; minimizing an energy of a system based on the first set of ansatz terms and the plurality of first amplitudes; computing a perturbation correction using one or more ansatz wave functions; determining whether the energy of the system is converged; and in response to determining that the energy of the system is not converged, predicting a second set of ansatz terms and a plurality of second amplitudes associated with the second set of ansatz terms.

Description

[0001] Cross-references to related applications

[0002] This application claims priority and benefit from the following patent applications: U.S. non-provisional application No. 17 / 177,813, filed February 17, 2021, entitled "METHODS AND APPARATUSES FOR RESOURCE-OPTIMIZED FERMIONIC LOCAL SIMULATION ON QUANTUM COMPUTER FOR QUANTUM CHEMISTRY"; U.S. provisional application No. 62 / 979,974, filed February 21, 2020, entitled "METHODS AND APPARATUSES FOR RESOURCE-OPTIMIZED FERMIONIC LOCAL SIMULATION ON QUANTUM COMPUTER FOR QUANTUM CHEMISTRY"; and U.S. provisional application No. 63 / 130,088, filed December 23, 2020, entitled "METHODS AND APPARATUSES FOR RESOURCE-OPTIMIZED FERMIONIC LOCAL SIMULATION". The contents of the U.S. provisional application for “ONQUANTUM COMPUTER FOR QUANTUM CHEMISTRY” are incorporated herein by reference in their entirety.

[0003] Government licensing rights

[0004] This invention was completed with government support from the DOE Basic Energy Science Foundation: DOE BES award de-sc0019449. The government holds certain rights to this invention. Technical Field

[0005] The various aspects of this disclosure generally relate to improving resource utilization in quantum computers. Background Technology

[0006] Atomic flux can be generated and used as neutral atoms or ion sources in certain systems. For example, some of these systems may include quantum information processing (QIP) systems. Trapped ions are one of the main implementations of QIP systems. Atom-based qubits can be used as quantum gates in quantum memories, quantum computers, and simulators, and as nodes in quantum communication networks. Atom-based qubits enjoy a rare combination of properties. For example, they exhibit very good coherence, can be fabricated and measured with near 100% efficiency, and are easily entangled with each other using suitable external control fields (e.g., optical or microwave fields). These properties make atom-based qubits attractive for extended quantum operations such as quantum computing or quantum simulation.

[0007] One application of quantum simulation is using QIP systems to simulate fermionic matter, including fermions undergoing local interactions. Simulation of fermionic systems on quantum computers can include two approaches: a variational quantum classical hybrid strategy suitable for imperfect, pre-fault-tolerant (pre-FT) quantum computers; and Hamiltonian dynamics simulations based on quantum simulation algorithms suitable for fault-tolerant (FT) quantum computers. In the context of ground-state energy estimation for fermionic systems, the former utilizes the efficient preparation of popular, well-preserved ansatz states and methods for evaluating the expected value of the Hamiltonian of the prepared ansatz states, both enabled by the quantum computer. In contrast, the latter leverages the ability of quantum computers to efficiently simulate quantum systems with local Hamiltonians, which, combined with quantum phase estimation, allows for the evaluation of the system's ground-state energy. In the pre-FT scheme, the quantum computation cost is likely dominated by multi-qubit gates. In contrast, in the FT scheme, circuits are typically written on a Clifford+T gate set, and the quantum computation cost is likely dominated by… Many of them are used for The Fourier Transform (FT) implementation of the gate. Therefore, improvements to the simulated quantum system may be desirable. Summary of the Invention

[0008] The following is a simplified overview of one or more aspects to provide a basic understanding of these aspects. This overview is not a comprehensive summary of all aspects considered, nor is it intended to identify key or necessary elements of all aspects, nor to depict the scope of any or all aspects. Its sole purpose is to present some concepts of one or more aspects in a simplified form as a prelude to the more detailed descriptions given later.

[0009] One aspect of this disclosure includes a method comprising: predicting a first set of ansatz terms and a plurality of first amplitudes associated with the first set of ansatz terms; minimizing the energy of a system based on the first set of ansatz terms and the plurality of first amplitudes; calculating a perturbation correction using one or more ansatz wavefunctions; determining whether the energy of the system converges; and predicting a second set of ansatz terms and a plurality of second amplitudes associated with the second set of ansatz terms in response to determining that the energy of the system does not converge.

[0010] Another aspect of this disclosure includes a quantum computing device configured to perform the following steps: predicting a first set of ansatz terms and a plurality of first amplitudes associated with the first set of ansatz terms; minimizing the energy of a system based on the first set of ansatz terms and the plurality of first amplitudes; calculating a perturbation correction using one or more ansatz wavefunctions; determining whether the energy of the system converges; and predicting a second set of ansatz terms and a plurality of second amplitudes associated with the second set of ansatz terms in response to determining that the energy of the system does not converge.

[0011] Aspects of this disclosure include: predicting a first set of ansatz terms and a first initial amplitude associated with the first set of ansatz terms; minimizing the energy of the system based on the first set of ansatz terms and the first initial amplitude; calculating the energy or at least one perturbation correction of the wavefunction based on one or more ansatz wavefunctions; determining whether the energy of the system converges; and predicting a second set of ansatz terms and a second initial amplitude associated with the second set of ansatz terms in response to determining that the energy of the system does not converge.

[0012] Aspects of this disclosure include a non-transitory computer-readable medium storing instructions therein that, when executed by a processor, cause the processor to: predict a first set of ansatz terms and a first initial amplitude associated with the first set of ansatz terms; minimize the energy of the system based on the first set of ansatz terms and the first initial amplitude; calculate the energy or at least one perturbation correction of the wavefunction based on one or more ansatz wavefunctions; determine whether the energy of the system converges; and, in response to determining that the energy of the system does not converge, predict a second set of ansatz terms and a second initial amplitude associated with the second set of ansatz terms.

[0013] Aspects of this disclosure include a system having a memory and a processor, configured to: predict a first set of ansatz terms and a first initial amplitude associated with the first set of ansatz terms; minimize the energy of the system based on the first set of ansatz terms and the first initial amplitude; calculate the energy or at least one perturbation correction of the wavefunction based on one or more ansatz wavefunctions; determine whether the energy of the system converges; and, in response to determining that the energy of the system does not converge, predict a second set of ansatz terms and a second initial amplitude associated with the second set of ansatz terms.

[0014] To achieve the foregoing and related objectives, one or more aspects include the features fully described below and specifically pointed out in the claims. The following description and drawings set forth certain illustrative features of one or more aspects in detail. However, these features represent only a few of the many ways in which the principles of each aspect can be employed, and this description is intended to include all such aspects and their equivalents. Attached Figure Description

[0015] The disclosed aspects will be described below with reference to the accompanying drawings, for illustrative purposes and not for limiting the scope of the disclosure, wherein similar reference numerals denote similar elements, and wherein:

[0016] Figure 1 This is a diagram illustrating an example of a standard two-body interactive circuit according to some aspects of this disclosure.

[0017] Figure 2 This is a diagram illustrating an example of an FT scheme optimization circuit for two-body terms according to some aspects of this disclosure.

[0018] Figure 3 The diagram shows a triple-controlled R with a relative phase triple-controlled NOT gate. x A diagram illustrating an example implementation of the (θ) gate.

[0019] Figure 4 The illustration shows examples of quantum circuits and optimized constructions based on some aspects of this disclosure.

[0020] Figure 5 This is a diagram illustrating another example of a circuit for optimizing the FT scheme for two-body terms according to some aspects of this disclosure.

[0021] Figure 6 This is a flowchart of a method for performing a simulation according to some aspects of this disclosure.

[0022] Figure 7 This is an example of a graph comparing the ground-state energies of water molecules at their equilibrium geometries, calculated by various methods using some aspects of this disclosure using the STO-3G basis set.

[0023] Figure 8 This is a circuit example of an adjacent Pauli string circuit according to some aspects of this disclosure.

[0024] Figure 9 The illustration is an example of a diagram that is partially improved according to some aspects of this disclosure.

[0025] Figure 10 The illustration shows an example of a histogram of an additional number of CNOT gates according to some aspects of this disclosure.

[0026] Figure 11 This is a block diagram illustrating an example of a quantum information processing (QIP) system according to some aspects of this disclosure.

[0027] Figure 12 This is a diagram illustrating an example of a computer device according to some aspects of this disclosure.

[0028] Figure 13 An example of a graph illustrating the reduction in the number of measurements according to some aspects of this disclosure is shown.

[0029] An appendix is ​​attached, the contents of which are incorporated in their entirety by way of quotation. Detailed Implementation

[0030] The detailed description that follows, taken in conjunction with the accompanying drawings, is intended as a description of various configurations and not as representing only the configurations in which the concepts described herein can be practiced. This detailed description includes specific details intended to provide a thorough understanding of the various concepts. However, it will be apparent to those skilled in the art that these concepts can be practiced without these specific details. In some cases, well-known components are shown in block diagram form to avoid obscuring these concepts.

[0031] As described above, trapped ions can be used to realize qubits used in quantum information processing (QIP) systems. A typical ion trap geometry or structure for quantum information and metrology purposes is a linear radio frequency (RF) Paul trap (also known as an RF trap or simply a Paul trap), in which nearby electrodes maintain static and dynamic potentials, resulting in the confinement of the ions. Qubits based on trapped ions can be used as various types of devices, including but not limited to quantum memories, quantum gates in quantum computers and simulators, and nodes in quantum communication networks.

[0032] As used in this disclosure, the terms "atom" and "neutral atom" are interchangeable, as are the terms "ion" and "atomic ion." Furthermore, the terms "atomic ion," "ion," "atom," and "neutral atom" can describe particles generated by the source material and which will be confined or trapped, or are actually confined or trapped in a trap to form a crystal, lattice, or similar arrangement or structure. The term "plasma" can describe the flux of neutral atoms, ions, or both.

[0033] One aspect of this disclosure optimizes the quantum circuitry for implementing fermion simulations in both pre-FT and FT schemes. The cost function considered includes the number of multi-qubit gates in the pre-FT scheme and the R0 required for application in the analog circuitry in the FT scheme. z Number of doors or R z The number of time steps for the gates. Another aspect of this disclosure includes product-formula (PF) based methods for FT schemes and variational methods based on unitary coupled cluster (UCC) ansatz for pre-FT schemes. In other aspects, fermion simulations may include methods using second-order perturbation-based approaches and the transformation from generalized fermions to qubit operators.

[0034] In some implementations, simulation results from the FT scheme can be used in chemical, pharmaceutical, physical, and other applications. The simulation can be real-time.

[0035] In some implementations, the ground state of a particle can be calculated using simulation results from a pre-FT scheme. The ground state of a particle can be used to determine chemical reactions, bonding properties, energy states, etc.

[0036] In one aspect of this disclosure, in the FT scheme, R z The (θ) and T gate counts can be optimized together with the auxiliary qubit count, assuming a product formula algorithm is used. In some cases, a 2-fold saving rate for the gate count and an 11-fold saving rate for the auxiliary qubit count can be achieved. In the pre-FT scheme, the number of two-qubit gate counts can be optimized.

[0037] One aspect of this disclosure proposes a framework that bootstraps a VQE process toward convergence to the ground-state energy of a fermionic system. This framework, based on perturbation theory, improves the energy estimation for each cycle of the VQE process. The improved energy approach can save the quantum resources (e.g., the number of qubits and / or quantum gates) required to reach a predetermined tolerance from a known ground-state energy.

[0038] Regarding the PF scheme, one aspect of this disclosure aims to reduce R z (θ) Number of gates and Rz (θ) gate depth, because R z (θ) gates can utilize more than one T gate, which is resource-intensive.

[0039] In one implementation, the fermionic system evolves according to the time-independent, local second quantization Hamiltonian H in the occupation basis.

[0040]

[0041] in and a q Let h represent the fermion production and annihilation operators at the p and q energy levels, respectively. pq and h pqrs Let represent the Hamiltonian coefficients of the single and double fermions, respectively. The fermion operators follow the anti-commutation relation of the gauge.

[0042]

[0043] Where {A, B} represents the anti-substitution AB+BA, δ jk It is Kronecker delta, and 1 is the identity operator. For implementation on a quantum computer, the fermion operator can be transformed using the Jordan-Wigner (JW) transform defined for an n-qubit system according to the following equation.

[0044]

[0045] Where j∈[0, n-1], Represents the tensor product.

[0046] To implement the evolution operator U on a quantum computer using quantum dynamic simulation methods evolution =e -iHt Where H is the system Hamiltonian, t is the duration of the system's forward evolution in time, and U is realized. evolution The algorithm can include PF techniques of order 2k, with a gate complexity of O(5^2). 2n t 1+1 / 2k / ∈ 1 / 2k 2k-order power phosphing (PF) techniques can produce more efficient quantum circuits in practice. They can be used to realize U... evolution Other algorithms include asymptotically optimal quantum signal processing, with a gate complexity of O(t+log(1 / ∈) / log log(1 / ε)), where ∈ is the algorithm's approximation error.

[0047] One aspect of this disclosure includes a quantum-classical hybrid approach based on variational quantum eigenvalue solver techniques. This technique includes implementation on a quantum computer. Where T is the clustering operator defined according to the UCC method. By adjusting the variational parameters in T, in a typical VQE method... The initial state is minimized, where |ψ0> is the initial state of the target ground state assumed to be close to H, and can be readily prepared on a quantum computer. Then, the goal of this hybrid approach is to estimate the ground state energy of the fermionic system using a Hamiltonian of the form shown in equation (1).

[0048] In an alternative implementation, the unitary ansatz evolution operator U can be implemented on a quantum computer (e.g., a unitarily coupled cluster ansatz). ansatz U can be adjusted. ansatz One or more variational parameters to minimize

[0049] In some respects, a complementary method to the above-mentioned hybrid approach based on perturbation theory can be used to estimate the ground state energy.

[0050] Figure 1 It is implemented as a diagram. An example of a block diagram of a two-body interactive circuit 100. In some implementations, by using σ + σ + σ - σ - (omitted for clarity) ) expands to σ x σ x σ x σ x , σ x σ x σ y σ y , σ x σ y σ y σ x , σ x σ y σ x σ y , σ y σ y σ x σ x , σ y σ x σ x σ y , σ y σ x σ y σ x , σ y σ y σ y σ yBy following a specific order and executing them sequentially, a two-body interactive circuit 100 can be obtained after applying well-known circuit optimization routines.

[0051] In some implementations, the 2k-order PF algorithm can be used to simulate fermionic systems. After performing a JW transform on the Hamiltonian H in equation (1), the time evolution operator can be written as follows.

[0052]

[0053] in where and

[0054]

[0055]

[0056]

[0057] in k > 1. The forms of each exponential term (hereinafter referred to as the Trotter term) are as follows: Where θ′ j It is an appropriately scaled θ j Standard circuits for implementing Trotter terms are well known in the art. Optimized quantum circuits can implement exemplary Trotter terms.

[0058] In some implementations, the two-body Trotter item Realize R between the |0011> state and the |1100> state X Rotation. For the input state |b0b1b2b3>, where the m-th qubit variable b m Including values ​​0 or 1, Map |0011> to |0111>, and map |1100> to |1111>. Triple control R for the zeroth qubit. X Rotation will achieve the desired rotation of the mapped state. Figure 2 An example of a quantum circuit 200 implementing a two-body term constructed according to the aforementioned method is shown.

[0059] Figure 2 Triple control R in X Gates can be implemented using single-qubit and / or two-qubit gates. For example, triple-control R... X Doors can be made with Hadamard, R z This is achieved using AND / or triple-controlled NOT gates. In one implementation, the triple-controlled R... z The gate can be a NOT gate with relative phase triple control, such as... Figure 3 As shown in (a). Figure 3 The circuit 300 shown in (a) will convert the R of each two-body Trotter term. z The count was reduced from the standard 8 to 2.

[0060] In some aspects of this disclosure, the aforementioned Trotter item can be used as follows: Figure 3 This is achieved using auxiliary qubits initialized with the |0> state shown in (b). Figure 3 (b) The circuit 350 shown may require one R for each two-body Trotter item. z Depth. For FT scheme fermion simulations, this construction may be the optimal R-value for the two-body Trotter term implementation. z depth.

[0061] In some implementations, triple control R x R of a gate (e.g., circuit 300) z Implementations optimized for T-gates include relative-phase Tooffoli gates, replacing traditional Tooffoli gates to reduce T-gate counts. In some respects, relative-phase Tooffoli gates with three controls may not require auxiliary qubits.

[0062] Circuit 300 includes two Rs configured in series. z To reduce R z The depth can be increased by introducing auxiliary qubits prepared for |0>. Triple control R is shown in circuit 350. x R of the door z An example of a deep-construction approach. In some cases, if the same angular rotation occurs simultaneously in multiple qubits (which is possible if the molecule of interest being simulated has symmetries), then a weight-sum method can be used to exponentially reduce R at a cost optimal for the number of qubits and T-gates. z The number of gates. Two R... to be applied in parallel. z Gates may not benefit from weights and methods. Figure 4 An example of a triple-controlled relative-phase Tooffoli gate 400 is shown. The triple-controlled relative-phase Tooffoli gate 400 is a logic NOT gate. Figure 4 Also shown is the singleton operator. Circuit 410. Figure 4 The R of the singleton term is further shown. x The deeply optimized circuit 420. Circuit 420 does not require auxiliary qubits or two R in one layer. z Door.

[0063] In some implementations, the above techniques can be generalized to a single entity. This generalization can be applied to any number of entities.

[0064] In some cases, if there exists a σ acting on some other qubits x，y，z Operators, for example, due to the JW transformation, can modify the two-body circuit as follows. For example, there exists a σ acting on an additional qubit. z Operator. In this case, in order to achieve Figure 2 Circuit 200 in the diagram can be modified to... Figure 5 The circuit shown is 500.

[0065] In some implementations, for singleton operators Similar to the two-body case shown above, we can use

[0066]

[0067] To implement this operator. z To achieve controlled R in the optimal way in terms of gate depth x The door, we passed through

[0068]

[0069] To implement this gate, when it is inserted into a single-unit operator circuit, it will generate R. z Depth-1 circuit.

[0070] In the pre-FT scheme, the ground state energy of the system, whose Hamiltonian is given by form (1), can be calculated, for example, by the variational quantum eigenvalue solver (VQE) method, where the energy of the ansatz state is evaluated by generating the ansatz state on a quantum computer. In the VQE method, the energy can be minimized (locally or globally) to obtain the ground state of the target system by iteratively calling the quantum computer to calculate the energy of the parameterized ansatz state.

[0071] In another aspect of this disclosure, one aspect of this disclosure can be used in pre-FT schemes for ground-state energy calculations. A type of ansatz state |Ψ ansatz >A parameterizable unitary ansatz evolution operator U can be used ansatz Transform from the initial state |Ψ0>, for example |Ψ ansatz >=U ansatz |Ψ0>。 Operator U ansatz It can be expressed in exponential form. Let Z be the denoted value. By changing the parameters in Z, the energy of the ansatz state is... It can be minimized using variational methods. Through appropriate fermion-to-qubit basis transformations, the energy expectation (the quantum resource cost is the operator U) can be efficiently evaluated on a quantum computer. ansatz (Implementation cost). The following process can be performed using the unitary evolution operator U. ansatz Any anasatz implementation. In one example, a unitarily coupled cluster anasatz with single and / or double excitation (UCCSD) can be used. Other examples include the k-product of the exponents of different pairs of coupled cluster double excitation operators (k-UpCCGSD) or a direct unitary extension of the anasatz of generalized unitarily coupled cluster (UCCGSD).

[0072] In some implementations, the method can begin with the ground-state Hartree-Fock (HF) wavefunction |Ψ0>, which can be computed on a classical computer and implemented on a quantum computer. Next, unitary operators can be used. Evolutionary wave function |Ψ0>, where It can be a cluster operator. Z 1，2 These can be single-excitation operators and double-excitation operators, respectively, and are written as follows in the second quantization:

[0073]

[0074]

[0075] Where t pr and t pqrs These are variational parameters, where virt and occ represent the virtual level and occupied level, respectively. To minimize the implementation of U... ansatz The size of the quantum circuit, which can lead to an ansatz state energy close to the ground state energy within a pre-specified tolerance, can take into account some or all terms in cluster operations. For example, in the VQE method, only all parameters t can be included. pr and t pqrs A subset of, the remaining parameters can be set to zero.

[0076] In some aspects of this disclosure, two processes can be used to reduce the circuit complexity of the methods detailed above, which is represented by the total number of multi-qubit gates. A first-order PF algorithm can be used to implement ansatz, although extensions to any higher-order PF algorithm are straightforward. Alternatively or additionally, a hybrid framework based on perturbation theory can also be used to reduce circuit complexity. In this hybrid framework, important ansatz terms can be contained within a larger instance of the VQE circuit based on a smaller instance of the VQE circuit, thus effectively guiding the VQE process to the ground-state energy estimation of the analog system.

[0077] In some implementations, the general framework can leverage the power of perturbation theory to optimize VQE-based quantum simulations by predicting the ansatz terms to be included in the UCCSDansatz and correcting the VQE results through post-processing. This framework can optimize the total number of VQE executions and the size of the ansatz state preparation circuit used to achieve convergence of the ground-state energy estimation. The derivation of a simple perturbation scheme that can be implemented directly within the framework can be termed a hybrid second-order perturbation. Plesset perturbation (HMP2) method.

[0078] In some aspects of this disclosure, any general (i.e., not specific to any system) unitary ansatz operator can be written as:

[0079]

[0080] The summation is performed on a set of orbital sequences {α}, f α It is the variational parameter set {t β The real function of}, and D α It is a general orbital replacement operator that replaces orbitals in a wavefunction based on an orbital sequence α. In one example, a single orbital replacement operator... Replace orbital q with p. For a specific general ansatz method, there is a complete set of variational parameters T. full ={t β}, which can be used to parameterize the ansatz evolution operator. In the case of UCCSD, T full ={t pr , t pqrs |p, q∈virt; r, s∈occ}, And f α =t α .

[0081] Next, a set of ansatz evolution operators can be generated. Each element is a unitary operator, which is passed through the complete variational parameters. subset T i Derive the ansatz state. Complement T full \T i At least some of the parameters can be fixed to an ordered set of default values. This default value can be absorbed into f. α In the middle, and the supplement T full \T i Elements in the array can be set to zero. For Each of them The energy of the ansatz state can be variationally minimized as Among the elements (T) i) represents set T i The elements. For a specific general ansatz in the pre-FT scheme via the VQE method, the goal is to find elements with reduced circuit complexity. Its satisfaction Where ε is the predetermined error standard.

[0082] In one aspect of this disclosure, convergence can be achieved with reduced circuit complexity by starting with an operator having a small set of variational parameters, which can achieve an error threshold and iteratively grow this set. From the m-th iteration to the (m+1)-th iteration, this set of variational parameters in the operator grows and satisfies One aspect of achieving fast convergence is the selection of additional variational parameters to be included between iterations. For example, one aspect of this disclosure may include, for a given anasatz state whose variational parameters have already been optimized (e.g., by a conventional method of VQE without perturbation energy correction), iteratively selecting the variational parameters to be included next in the preparation of the anasatz state based on the magnitude of the perturbationally predicted wavefunction correction amplitude.

[0083] Figure 6 A flowchart of the proposed method 600 is illustrated. Method 600 may include a systematic approach that uses perturbation methods to improve the quantum resources required for the VQE method. Specifically, method 600 can enable the system to converge rapidly to the ground state energy of the quantum circuit. Rapid convergence may depend on the choice of a single excitation term in equation (8) used to prepare the subset of variational parameters in the UCCSD example.

[0084] In one illustrative embodiment, similar to the conventional VQE method used for fermion simulations, this method begins with the ground-state Hartree-Fock wavefunction of the single-particle Hamiltonian as |Ψ0>. Method 600 can be implemented by a classical computer, a quantum computer, or a hybrid classical-quantum computer. In one aspect, method 600 can be implemented by computer device 1200, as described below ( Figure 12 ).

[0085] In box 605, method 600 can predict initial guesses of the ansatz terms and associated magnitudes of the initial group. For example, the first step may include determining the initial evolution operator. This is used for the first iteration of VQE. In one implementation, a two-particle Hamiltonian is used as the perturbation and energy, and the wavefunction correction can be calculated using classical algorithms. By perturbing the wavefunction, the amplitudes of each excited state can be extracted from the single-particle Hamiltonian basis. These amplitudes can serve as initial guesses for the variational parameters of the first round of VQE simulations, which can significantly reduce the number of evaluations of the quantum circuit compared to all-zero or random initial guesses. If the energy convergence criterion is δ... E(That is, when the magnitude of the energy change associated with the addition of any additional ansatz term is less than δ) E When the entire simulation is considered convergent, the initial ansatz set can include contributions to the relevant energy greater than f(δ). E The ansatz term of ) where the function f(δ) E The standard selection can simply be δ. E .

[0086] In one aspect of this disclosure, method 600 may include selecting a perturbation method, such as a second-order perturbation method. -Plesset (MP2) perturbation theory. Method 600 may include selecting a set of ansatz terms. In one implementation, this set of ansatz terms may be compatible with unitary types later used on a quantum computer. An example includes UCCSD. The set may contain a finite number of ansatz terms, each of which is distinct. Method 600 may execute a selected perturbation method on a classical computer using the ansatz terms from the selected set. This execution may result in an initial perturbation strength associated with each ansatz term for use by the quantum computer.

[0087] In box 610, method 600 can use the ansatz terms of the initial group and their initial variational parameter values ​​to perform the first round of VQE simulation to minimize energy. In some cases, additional measurement terms can be added to the mixed perturbation step.

[0088] In box 615, once the energy is minimized and the ansatz state converges, method 600 can use the ansatz wavefunction to calculate the energy and / or the perturbation correction of the wavefunction. For example, method 600 can calculate the correction to the correlated energy and the wavefunction. The final total energy of this period can be the sum of the energy correction and the VQE energy. The details of the hybrid perturbation method are discussed in more detail below.

[0089] In some aspects of this disclosure, equipped with an ansatz term and initial perturbation strength determined by performing a selected perturbation method using an ansatz of a selected group on a classical computer, method 600 can create a VQE quantum circuit that can generate an ansatz state for a popular ground state. Based on the Hamiltonian of the system to be simulated, method 600 can attach the required measurement-based transformation circuitry for each term in the Hamiltonian to determine its expected value. In the case of hybrid methods such as HMP2, method 600 can additionally consider an energy correction operator, which often results in the need to consider additional measurement-based transformation circuitry. In some embodiments, method 600 can compile and / or optimize the circuitry and run the resulting circuitry on a quantum computer to evaluate the expected value of the Hamiltonian. In hybrid cases, method 600 can additionally or alternatively evaluate the expected value of the energy correction term on the quantum computer. Method 600 can use a classical optimizer to feed forward new variational parameters for experimentation, which can replace the initial perturbation strength used above. The above process can be repeated until the energy estimate converges to a minimum.

[0090] In box 620, method 600 can determine whether the energy calculation has converged.

[0091] In box 625, method 600 can predict the initial guesses for the next ansatz operator to be used and the newly added variational parameters based on the ability of the new operator to recover the wavefunction correction. For example, before starting the next cycle, method 600 can optionally identify a pool of operators that can be direct successors of the current operator. Next, method 600 can evaluate how much each operator in the pool can contribute to the perturbation wavefunction correction. The operator that best recovers the wavefunction correction can be used in the next cycle (if any), and the way the correction is recovered can indicate the initial guesses for the newly added variational parameters in the next cycle. The initial variational parameter values ​​for the ansatz term, which continues as part of the next VQE simulation, can be imported from the previous cycle.

[0092] In some implementations, if the energy estimate based on the selected ansatz is still far from the actual ground-state energy, or in the absence of a known ground-state energy, method 600 can make an educated guess by systematically increasing the ansatz magnitude (described below). Method 600 may include determining whether further increasing the ansatz significantly alters the energy estimate. Method 600 may include determining which ansatz term to add to the existing ansatz based on a hybrid perturbation framework. An initial guess of the perturbation strength associated with the new term is also determined based on the hybrid framework.

[0093] In block 630, method 600 can output the simulation results.

[0094] On one hand, the techniques described in method 600 can be executed by a classical computer to generate simulation parameters (e.g., ansatz terms) for executing quantum circuits on a quantum computer. On the other hand, the techniques described in method 600 can be executed on a hybrid classical-quantum computer to generate simulation parameters for executing quantum circuits on a hybrid classical-quantum computer.

[0095] This cycle of iterations encapsulates the process of running VQE simulations and calculating hybrid perturbations until the total energy converges (in box 630).

[0096] The Plesset perturbation method can be mixed with VQE simulation, hereinafter referred to as HMP2. The HMP2 method improves the energy of each VQE cycle by correcting the perturbation energy and optimizing the ansatz operator to be used in the next cycle (if any) through perturbation wavefunction correction. In the m-th VQE cycle, the energy of the ansatz state... (In the context of first-order perturbation theory) it can be written as:

[0097]

[0098] Where F is the Fock operator, E0 is the sum of orbital energies, and E (0) =E0 is the zeroth order energy. This is the first-order correction energy. Based on perturbation theory, the second-order correction energy can be written as...

[0099]

[0100] in The set of orbital sequences {α′} can be the same as or different from the set {α}. When {α} is a finite set, the set {α′} can include the set {α}. Energy This could be the orbital energy difference that substitutes for the orbital energy. In one implementation, Where E p and E q These can be the orbital energies of the p-th and q-th orbits, respectively. Substituting into equation (18), the numerator of equation (18) becomes

[0101]

[0102]

[0103] We used

[0104] In order to convert VQE results (e.g., ansatz wavefunction) Directly applied to perturbation calculations, substitute the following terms into equation (19): in and To obtain

[0105]

[0106] Next, Taylor can be expanded and applied. In until The first order in the middle obtains

[0107]

[0108] The energy correction for VQE can be obtained using equations (18) and (21).

[0109] Terms in the appropriate transformation space from fermions to qubit bases It is the Pauli operator The sum of the products. Because It has an eigenvalue of +1 or -1, which represents and As eigenvectors with their own eigenvalues, equation (21) can be rewritten as follows:

[0110]

[0111]

[0112] In some implementations, equation (22) can utilize |Ψ ansatz In or The projection onto the surface. Second-order correction energy can be obtained within this circuit size without any quantum resource overhead. Wavefunction correction can be used to optimize the selection of the ansatz evolution operator in the next cycle. The ansatz state wavefunction can be considered as the zeroth-order wavefunction |Ψ. (0) Furthermore, the first-order wavefunction correction can be given by the following equation.

[0113]

[0114] In some respects, the following aspects of this disclosure can be used to determine which ansatz evolution operator to use in the (m+1)th round. The variational determination set T is performed in the m-th round. m The set of parameter values ​​in (denoted as S) m After that, the method can search for the satisfying and n(T) i \T m ) = min j≠m [n(Tj \T m Evolution operators of )] set Where n(A) represents the number of elements in set A. R m This can be a pool of operators that can be selected for the next ansatz evolution operator. For R m Each of them Overlap can be calculated as in

[0115]

[0116] Identity can be used Estimate based on equations (18), (21) and (24) Straight. From R m The one selected with the largest of It can be used as the ansatz evolution operator for the next cycle. In some implementations, wavefunction correction can be used to apply the additional variational parameter t. β ∈T m+1 \T m The initial value is guessed as

[0117] In the case of UCCSD, R m Each of them exist It may be more than In One more D α This corresponds to the incremental change of the terms contained in the ansatz operator Z. Using It can be approximated by the corresponding additional D α Disturbance amplitude

[0118] In some respects, the process described above for the m-th cycle of the VQE cycle can also be used by taking m=0 before the first cycle. In this case, the calculation can be classical, and the energy correction can be MP2 energy correction. The wavefunction correction can inform the first operator to be tried and suggest initial values ​​for the corresponding variational parameters.

[0119] The number of Pauli string operators whose expected values ​​will be evaluated using quantum computing scales (e.g., in equation (22)) is O(n^2) for each UCCSD ansatz. 8 ) where n is the number of qubits. Here, for a fixed ansatz, Z can be fixed, and equation (21) may require O(n8 ) Pauli string measurements, O(n 4 ) from In addition, O(n 4 ( ) from H. While this metric may seem challenging, in practice, some aspects of this disclosure can be applied to reduce the number of evaluations, as described below.

[0120] One aspect of this disclosure involves the selection of qubits. Since only qubits coupled via selected ansatz are entangled, some qubits can be used to generate ansatz states, while others can operate in a classical manner. On the other hand, extensions of small, disjoint sets of qubits with a set size that allows for inexpensive classical post-processing can be utilized. In some embodiments, if two qubits represent the same spatial orbital with two opposite spins, and their energies are indistinguishable due to the specific selection of ansatz, these two qubits can encode redundant information. This allows information to be encoded using only a single qubit.

[0121] One aspect of this disclosure includes simultaneously and / or in parallel measuring multiple Pauli strings to reduce the total number of measurements, provided that the Pauli strings are commutable to each other. Methods of measurement may include general commuting (GC) partitioning and qubit-wise commuting (QWC) partitioning. GC partitioning can reduce the number of measurements but incurs the additional cost of additional two-qubit gates. QWC partitioning can reduce the number of measurements (to a level lower than GC partitioning) without increasing the number of two-qubit gates. In some aspects, QSR can be performed before GC to reduce the additional two-qubit gates.

[0122] Figure 7 An example of graph 700 is illustrated, which shows the effect of D included in the UCCSDansatz operator. α The convergence of ground-state energies using different methods increases with the number of terms N. The insertion order of the ansatz terms in the conventional curves shown here is obtained from prior knowledge of the contribution order of the determinant in classical FCI calculations, which closely resembles the ideal case but is difficult to obtain in practice. Comparing the two convergence curves, the HMP2-assisted simulation captures the dominant ansatz terms. Figure 7The good consistency between FCI ordering and HMP2 ordering is shown. HMP2 correction of the ground state energy can help accelerate energy convergence towards the FCI energy. The fast energy convergence achieved by the HMP2 method can be beneficial for quantum computers, given that the implementation of each additional anasatz term leads to significant noise accumulation in noisy intermedium-scale quantum (NISQ) hardware.

[0123] In some cases, various fermion-to-qubit transformations can be used to reduce the quantum resource requirements of pre-FT fermion simulations. Well-known fermion-to-qubit transformations, such as the Jordan-Wigner (JW) or Bravyi-Kitaev (BK) transformations, map fermion-generating (annihilating) operators to Pauli strings. However, in principle, many other transformations are available, such as generalized transformation (GT) methods, of which the JW and BK transformations are part. When the PF algorithm is used as a concrete example of implementing UCCSD ansatz, significant quantum resource savings can be achieved through appropriate selection of the mapping to the given cluster operator inputs and a carefully chosen sequence of heuristic optimization methods.

[0124] The transformations in the GT method can follow the relations specified in equation (2). This can be achieved by considering the following invertible upper triangular basis transformation matrix β, which transforms the occupied basis into the GT basis according to the following equation.

[0125]

[0126] Where β i，j ∈{0, 1}, i > j, β i，j = 1, where n is the number of qubits involved in the transformation. Matrix operations are performed in modulo-2 space, and the main diagonal elements are excluded when generating the set. For convenience, the following index set is defined.

[0127] • Update set U(j): The elements of this set are the row indices of the j-th column of the basis transformation matrix β that have non-zero elements.

[0128] • Flipped set F(j): The elements of this set are the inverses of the basis transformation matrix β. -1 The column index of row j containing non-zero elements.

[0129] • Parity check set F(j): The elements of this set are matrices (πβ) -1 -β -1 The column index of row j containing non-zero elements, where π i，j 1. If i ≤ j, then π i，j1, otherwise 0.

[0130] • Remainder set R(j): The elements of this set are matrices πβ -1 The column index of row j containing non-zero elements.

[0131] Then, based on the GT generation and annihilation operators,

[0132]

[0133]

[0134] It can be shown to satisfy equation (2). The JW transformation is a special case of β = 1.

[0135] In some implementations, the UCCSD ansatz implementation relies on the JW transform, and it considers a carefully selected set of heuristics to optimize the resulting quantum circuit. Details of the heuristics considered are discussed below. An overview of the steps in the application sequence is given below.

[0136] The outermost loop of the method discussed above considers different transformation matrices β. For a given mapping matrix β, the following routines can be executed to construct and optimize circuits implementing UCCSD ansatz. These routines are designed to repeatedly invoke a dedicated set of automated circuit optimization tools. The efficiency of these tools allows for rapid evaluation of the cost function for different cases in each subroutine.

[0137] In one implementation, the first routine can be associated with a Fermi level label. Unlike the JW transformation where U(j) is an empty set and P(j) = R(j), in the GT method, an infinite number of combinations of sets U(j), P(j), and R(j) are possible. To take advantage of this, careful selection of which Fermi level to map to which qubit index might be necessary. However, exploring all possible mappings is computationally very expensive. Instead, a greedy approach can be used where the routine explores one permutation at a time from a given Fermi level to a qubit index mapping. Specifically, from that given mapping, the permutation that results in the greatest reduction in quantum resource requirement is applied. This process can be iterated until no single permutation results in a reduction in quantum resource requirement. The following subroutine illustrates the cost function evaluation for each permutation.

[0138] In some respects, the first routine can reduce the number of two-qubit gates by utilizing different mappings from Fermi level labels to qubit indices. In applying the chosen... After the bit transformation, the Pauli string of the excitation operator depends on the qubit index value. An optimized mapping from Fermi level labels to qubit indices requires a smaller number of two-qubit gates. This mapping may be equivalent to maintaining a simple mapping from any Fermi level k to qubit index k while rearranging the Fermi level labels (which can be arbitrary). The degrees of freedom in Fermi level labeling can be used to reduce the number of two-qubit gates needed to implement single-body and / or two-body Trotter terms.

[0139] In some respects, this greedy approach can be implemented for an n-qubit system representing n Fermi levels. First, a unique set of permutation matrices {P} can be generated for possible k-swap operations. i Here, the k-swap operation can include swapping k unique ordered index pairs (2l... j ,2l j +1) and k other unique ordered index pairs with the same index j∈{0,...,k-1} (2m j 2m j +1) Swap, where and Example values ​​for k can be 2, 5, 10, 20, etc. Next, by representing the initial fermion labels as column vectors L0, one aspect of this disclosure may include traversing some or all of the permutation matrices in the first round, and calculating the number of two-qubit gates for each excitation term obtained from the subroutine rearrangement of the intra-Trotter term discussed later, along with the relabeled Fermi level P. i The number of two-qubit gates corresponding to each instance of L0. When traversing the permutation matrix, if any matrix results in a two-qubit gate count lower than any or all previous matrices, this two-qubit gate count can be recorded as N1, and the corresponding relabeled energy level vector is recorded as L1. Next, one aspect of this disclosure can iteratively perform this greedy method after the first round. The resulting relabeled Fermi level vector from the m-th round is represented as L. m And the corresponding two-qubit gate count is represented as N. m One aspect of this disclosure may include traversing some or all of the permutation matrices and targeting the relabeled energy level P. i L m For each instance, calculate the corresponding two-qubit gate count. Initially, take N. m+1 =N m When the matrix causes more than N m+1 When counting low-level two-qubit qubits, the gate count is recorded as N. m+1 And record the matrix as L m+1 If no matrix causes a difference greater than N... mA low two-qubit count can terminate the iteration and can reduce L m Return the result as the best one. Otherwise, proceed to the next round of the routine.

[0140] In some cases, once the labels are determined, they are applied to the relevant fermionic operators, including the molecular Hamiltonian, to maintain consistency during the simulation.

[0141] In different implementations, the second routine can be associated with the ordering of inter-trotter terms. Properly ordering trotter terms can significantly reduce the two-qubit gate count due to gate cancellation between adjacent trotter term circuits. While similar approaches are possible, a nontrivial modification can be used in the GT method. Specifically, each trotter term can be processed to determine whether it qualifies for classification into the same equivalence class, whose elements have the opportunity to save resources when placed adjacent to each other on the quantum circuit via the aforementioned gate cancellation. Once the equivalence class based on the eligibility of each trotter term is determined, a simple greedy method can be used to order the trotter term elements of each equivalence class to reduce the two-qubit gate count.

[0142] In some aspects of this disclosure, subroutines can be associated with the ordering of inter-Trotter terms. In this subroutine (also called the second routine), heuristics can be used to reduce the number of two-qubit gates by leveraging the degrees of freedom of arbitrarily ordered Trotter terms. Any ordering follows the error bounds provided by the PF algorithm. In one aspect of this disclosure, a preprocessing step is run based on information derived from the intra-Trotter term optimization step described below. Specifically, for the operator... and The subroutine checks whether any of the indices i and j of the former and i, j, k, and l of the latter can or cannot be used as target qubits in the circuit implementation of the operator in the standard compilation. Qubit indices that cannot be used as targets are marked as unqualified.

[0143] In one scenario, after determining that an item is unqualified, the subroutine continues using a greedy approach. The subroutine identifies the most frequent qualified index (e.g., p) among all items. Items with index p are then grouped into qualified target qubits and classified under equivalence class [p]. Next, all group elements in the lists of one-body and two-body operators are removed. This process is repeated until no operators remain in the lists.

[0144] In some aspects of this disclosure, quantum resource cost reduction can be achieved between element circuit representations of the same equivalence class, since the target qubit of the two-qubit gate is identical in each element of the same class. Once some or all equivalence classes are specified, the subroutine permutes the order in which the elements are implemented on the quantum circuit. Since considering all permutations can be very expensive, the subroutine can implement a greedy approach by starting with the two elements that result in the maximum resource cost reduction. Next, based on the reduction in resource cost, the subroutine connects the next element identified from the set of elements not yet implemented in the circuit. The connection process is repeated until there are no more elements in the set. In some cases, testing which element might be optimal for a given iteration may involve four cases per trial. Circuit connections can be performed as prefixes or suffixes, and the elements to be connected can be considered in their original intra-term order or in reverse order.

[0145] In some cases, subroutines may be associated with the ordering of Intra-Trotter terms. For a given mapping of fermion tags to qubit indices, efficient methods for implementing single- or double-fermion excitation trotter terms can be achieved using the JW transform. These methods may include optimizing circuit structures that utilize full qubit-to-qubit connections. The method relies on careful ordering of implementations of the intra-Trotter operator, e.g., exemplary double-fermion trotter terms. Extended to σ-based x，y，z The intra-Trotter term. In order to be efficiently implemented in other transformations used in our GT method, the cost function for each possible permutation of the intra-Trotter term can be computed.

[0146] On the one hand, the intra-Trotter subroutine can reduce the number of two-qubit gates by optimizing the ordering of the Pauli strings derived from the two-body operator. Each single-body term may result in an ordering up to the reverse, as it contains two Pauli strings. Therefore, a single-body term does not require a specific ordering. For the two-body operator, after applying the appropriate transformation and PF algorithm, it may contain eight sub-terms:

[0147]

[0148] Where j represents the intra-Trotter term index, v represents the qubit index, and σ∈{1, σ x , σ y , σ z Each intra-Trotter item All can be converted into standard circuits. In one implementation of U using arbitrary fermion-to-qubit transformations, each intra-Trotter term can lead to 2(N) j -1) CNOT gates, where N j It is the non-identical σ in the j-th Pauli string. (j，v) The quantity. In the JW transform, any choice of t, as long as σ (j，t) Either σ x Either σ y That is, qubits corresponding to the fermion tags in the two-body term are all good choices. If σ is in any of the eight Pauli strings in the above U... (j，t) If σ = 1, then the expected target qubit index t may be unavailable. Therefore, in the GT method, this leads to σ... (j，t) The qubit selection t with 1 can be removed from the target set that starts with all qubits that correspond to the fermion tags that appear in the two-body term of interest and will not subsequently be used in the inter-Trotter term rearrangement subroutine described above.

[0149] In some implementations, for a selected target qubit of choice t, the reordering of intra-Trotter terms can maximize the reduction in CNOT (e.g., the reduction in CNOT between adjacent Pauli string circuits). For adjacent intra-Trotter terms (i.e.... and For each of j ∈ [0, 6], v can be enumerated by all non-target qubits. By comparing σ (j，v) and σ (j+1，v) For a specific control qubit v, if the two Pauli matrices σ (j，v) and σ (j+1，v) If none of them are 1, then the circuit can be expressed as follows: Figure 8 The circuit 800 is shown. In circuit 800, t represents the target qubit, v represents the control qubit, and σ... l ∈{σ x , σ y , σ z}and If σ l =σ x Then M l =H. If σ l =σ y ,but If σ l =σ z Then M l =1.

[0150] In some cases, within circuit 900, if M0 = M2, there are two CNOT reductions. If M0 ≠ M2, there is one CNOT reduction. Assume that for the selected target qubit t, we have... A reduction in dual CNOT and The number of CNOTs reduced is [number]. Therefore, for the selected target qubit t, the total number of CNOTs reduced is [number]. For a selected objective t in a given sort, the total number of optimizations in CNOT is:

[0151] The resulting optimized circuit implements UCCSD ansatz in the selected transform basis defined by β. However, apart from the JW transform with β=1, the GTβ matrix requires initial mapping of the basis at the start of the circuit. This introduces overhead in two-qubit gate counting. To obtain the final quantum resource requirements, an automatic optimizer can be invoked using an input quantum circuit, which includes a prefix sub-circuit implementing β and a suffix sub-circuit implementing β-based UCCSD ansatz.

[0152] In practice, in the JW transform, whenever a pair of adjacent qubits (regardless of which appropriate Fermi level they correspond to) is excited to another pair of adjacent qubits representing another set of Fermi levels, the circuit implementing this excitation term can sometimes be simplified to require only two two-qubit gates, while requiring only half the number of qubits originally needed. To take advantage of this, a juxtaposition of a boson circuit written according to the JW transform and a non-boson circuit written according to the GT method can be used. Returning from the half-qubit space of the boson circuit to the full-qubit space of the non-boson circuit may require n / 2 CNOT gates.

[0153] In some implementations, a correspondence can be established between generalized boson terms and fermion terms. A fermion double excitation term. After being transformed by the JW transform, it becomes If p and q belong to the same spatial orbit, and r and s belong to another spatial orbit, assuming no other terms in the circuit violate the symmetry between p and q or r and s, then the p and q energy levels can be encoded by a single qubit, and r and s can also be encoded by a single qubit. Therefore, using p and r as representatives, the qubit spatial operator can be simplified to... Where k′ operates on a set of qubits, σ z Operators need to be applied to a given excitation term in a properly shrunk space. A properly shrunk space can include a single qubit representing each shrunk symmetric energy level and an unshrunk qubit. For example, if two energy levels k1 and k2 in the original space require σ... zAnd if k1 and k2 are encoded into a single index k′, then It can be called twice (once for k1 and once for k2). This is likely to be identical, thus saving resources.

[0154] Table II shows the circuit metrics for the optimal GT transforms found for JW, BK, and the heuristic toolchain specified above. These metrics are measured based on the number of two-qubit gates used to implement the UCCSD ansatz circuit for different molecules. Particle swarm optimization can be used to find the optimal GT transform. The advantage provided by the GT transform varies depending on the suite of molecules, ranging from 1.44% to 21.43%. This demonstrates the capability of the heuristic approach—that it is indeed possible to further optimize the quantum circuits obtained via the JW transform by considering the selection of a customized GT transform for different input cases. The low reduction rate is likely due to the inability of classical optimization to sample a sufficient number of matrices using limited classical computational resources.

[0155]

[0156]

[0157] In some examples, it is possible to use R in quantum simulation circuits z In gate depth reduction, parallel implementation of multiple Trotter terms is considered. Based on the circuit construction discussed above, methods can be considered to optimize this parallel implementation.

[0158] As mentioned above, please note that the circuit implementing the Trotter term mainly consists of three parts: an initial CNOT gate network for calculating a linear function of the Boolean input variables, a triple-controlled R... x The inverse of the initial CNOT gate network. Representing the Boolean variables at the input as a, b, c, and d in top-to-bottom order, the CNOT gate network outputs... and The following three outputs are in triple control R x The door remains unchanged. This means... and Any linear function can be used to implement JWσ with other Trotter terms. z The corresponding CNOT gates are linked. This is because, according to... Figure 4 The circuit structure shown requires invariance to achieve the proper inverse of the CNOT gate, which is first used to consider JWσ. z The heuristics for collecting these Trotter terms and implementing them simultaneously can then be used to optimize the depth of quantum circuits.

[0159] For pre-FT scheme VQE simulations, one aspect of this disclosure includes a general framework utilizing the predictive and corrective capabilities of perturbation theory. In another aspect, in addition to the HMP2 method described herein, many more complex forms of perturbation theory can be used to improve the efficiency and accuracy of the simulation. For example, the coefficients of ansatz terms, which may include triplet or higher excitations, can be perturbatively obtained in a manner similar to the classical CCSD(T) or CCSDT-1a / b methods that include triplet excitations. The above methods are general enough that other perturbation methods can replace the HMP2 method in perturbation-assisted quantum simulation cycles.

[0160] In some implementations, when the framework for considering bosonic terms in the JW transform and non-bosonic terms in the GT transform is extended and generalized, using multiple fermion-to-qubit transformations β for a given set of excitation terms may be valuable in reducing overall resource requirements. Given that different β transforms result in different resource requirements for the excitation terms in realizing the target UCCSD ansatz circuit, it is anticipated that a subset of excitation terms can be realized more efficiently by one β transform, and some other excitation terms by another. Therefore, dividing the set of excitation terms required to prepare the UCCSD ansatz state into several subsets of excitation terms, and realizing these subsets more efficiently by selecting appropriate β transform options for each subset, can prove more advantageous in terms of quantum resource requirements. Without departing from the spirit of this application, the anticipated trade-off between the indirect costs arising from switching transforms and the savings gained through customized selection of transforms can be optimized.

[0161] In another aspect of this disclosure, a process for optimizing the binary matrix β for the ground truth can be implemented using binary particle swarm optimization (PSO). To encode this problem, the upper triangular elements of the n×n binary matrix β can be mapped to a one-dimensional binary vector X∈{0,1}. d , where size Vector X can be used as the position vector of the PSO. The cost function of the PSO at position X mapped from the β matrix can be defined as the number of two-qubit gates required to implement the UCCSD evolution operator using a GT with a β matrix and a subsequent series of heuristics.

[0162] One aspect of this disclosure begins with the creation of a particle swarm, each particle having an initial position X mapped from the β matrix. i (t=0), the β matrix is ​​sampled from a set of upper triangular matrices where diagonal elements are 1 and off-diagonal elements are zero except for k of them. The non-negative integer t represents the time step, and t=0 can be the start of optimization. The total number of particles in the swarm can be... in This represents the binomial coefficient. Specifically, for n≤8, k can range from 1 to k. max =6 changes, for the other cases k max =3. Each particle can have the same properties as X. i (t) Velocity vector V of the same dimension i (t). The initial matrix elements of the velocity vector can be set to zero.

[0163] At any or every time step t, the process can track the local optimal position vector V of each particle i. i (t) and the corresponding local optimal cost s i (t) and the global optimal position vector G(t). The local optimal position vector L i (t) can be defined as having the lowest cost function value s at all positions where particle i has traveled to time step t. i The position vector of s(t). The global position vector is the position vector of all particles. i (t) Minimum L i (t). The velocity vector is updated from step t to step (t+1) according to the following formula:

[0164]

[0165] Where w is the inertia parameter, c1 is the cognitive parameter, and c2 is the social parameter. In some examples, various parameters can be used, such as w∈[-4, 4], c1∈[0, 2], and c2∈[0, 2]. The subscript j indicates the j-th element of the corresponding vector. If Then position vector The elements are updated to 0 if they are not in the range [0.0, 1.0], and to 1 otherwise. Here, rand() can be a pseudo-random number generator that samples from a uniform distribution on [0.0, 1.0].

[0166] Optimization can continue until t reaches t. max Or each position vector oscillates between two vectors for more than Then, the globally optimal position vector is used to map back to the optimal β matrix. In one example, For n≤8, t max =10000, for other cases, t max =100. In other examples, t max The choice can be large enough that the optimization reaches t. max It started oscillating earlier.

[0167] An example of the GT results using binary PSO can be represented based on a partial improvement of the two-qubit gate counting ρ, defined as follows:

[0168]

[0169] Where f GT and f JW This can be the number of two-qubit gates, used respectively to implement the UCCSD evolution operator using GT or JW with binary PSO. GT It depends on the classical computational resources used in binary PSO, which are primarily determined by the number of particles used, because t max It can be set large enough. To compare the amount of classical computational resources used with the maximum amount that might be needed, a fractional classical computational resource metric is the ratio between the number of particles N used and the total number of possible variations of the β matrix, which can be written as:

[0170]

[0171] Where n is the number of diagonal elements of the β matrix. Since the number of particles increases with increasing k, as mentioned above, k can be changed to effectively alter the amount of classical computational resources used for binary PSO.

[0172] In one aspect of this disclosure, circuit optimization techniques may include implementing a first routine associated with the Fermi level label.

[0173] In different respects, circuit optimization techniques may include implementing subroutines associated with the ordering of inter-Trotter items.

[0174] In some respects, circuit optimization techniques may include implementing subroutines associated with the sorting of intra-Trotter items.

[0175] In some aspects of this disclosure, one or more optimization techniques may be implemented. For example, circuit optimization techniques may include implementing a Fermi level marking routine and an inter-Trotter term ordering subroutine. In another example, circuit optimization techniques may include implementing a Fermi level marking routine and an intra-Trotter term ordering subroutine. In yet another example, circuit optimization techniques may include implementing an inter-Trotter term ordering subroutine and an intra-Trotter term ordering subroutine. In some examples, circuit optimization techniques may include implementing a Fermi level marking routine, an inter-Trotter term ordering subroutine, and an intra-Trotter term ordering subroutine.

[0176] Figure 9Simulation plots 900, 910, 920, and 930 are shown as fractional improved ρ functions of the fractional classical computational resource R used as a binary PSO. The first simulation plot 900 is associated with hydrogen fluoride (HF). The second simulation plot 910 is associated with beryllium hydride (BeH2). The third simulation plot 920 is associated with water (H2O). The fourth simulation plot 930 is associated with ammonia (NH3). The simulation plots can use different numbers of ansatz terms (e.g., 4–6 ansatz terms). Simulations were performed using the HMP2 method with UCCSD ansatz, via the STO-3G basis set. For HF, BeH2, and NH3, the final UCCSD ansatz operators contain 3, 9, and 52 excitation terms, respectively. This allows for estimation of the corresponding ground-state energies within the chemical accuracy range from their known ground-state energies. When the term n is sufficiently small to explore a relatively large portion of the possible variations in the β matrix (R > 10), the results are considered satisfactory. -5 The improvement ranges from less than 14% to more than 20%. A larger n may result in less improvement.

[0177] In another aspect of this disclosure, the process for qubit space reduction (QSR) techniques can be used to optimize the number of Pauli strings to be measured. In a first variant, the process can classically handle qubits in classically accessible states. In a second variant, the process can utilize boson states by expanding only a single qubit of two spin orbitals in the same spatial orbital, and possessing degeneracy energies due to a specific choice of ansatz.

[0178] For the first variant, in some cases, consider the Pauli string to be measured, S = A0A1...A n-1 A i ∈{σ x , σ y , σ z , I}, and I is a two-by-two identity operator. Taking the above conjugate with quantum states ||Ψ>=|ψ>|φ>, where |ψ> represents the quantum state of a qubit entangled by ansatz, and |φ> represents the quantum state of a qubit in a classically accessible state (e.g., the computational fundamental state), we can write the following equation:

[0179]

[0180] Here, sets P and Q represent the set of entangled qubits and the set of qubits in a classically accessible state, respectively. The second tensor product in equation (30) can be computed classically and / or efficiently. Therefore, only the first tensor product in equation (30) may require a quantum computer, which leads to a reduction in the number of Pauli strings to be measured.

[0181] The following describes an implementation example of the second variant. The quantum state |ψ> can be defined as:

[0182] |Ψ>=∑ m |ψ m >(c m，0 |00>+c m，1 |11>). (31)

[0183] Since the separated two-qubit states |00> and |11> only encode the information of a single qubit, equation (31) can be compressed to:

[0184] |Ψ comp >=∑ m |ψ m >(c m，0 |0>+c m，1 |1>) (32)

[0185] Pauli string S=A o A1...A n-1 It can exist in the complete space spanned by |Ψ> in equation (31). Here, A0A1 can be the Pauli product located in the two-qubit state space separated in equation (31). Conjugating S and |Ψ>, it can be written as the following equation.

[0186]

[0187] Compressed Pauli string S comp =A comp A2...A n-1 It can exist in equation (32) by |Ψ comp >Spanning the entire space. Conjugate S and |Ψ>, which can be written as the following equation.

[0188]

[0189] Based on equations (33) and (34), the two equations can be consistent if the following conditions are met.

[0190] <00|A0A1|00>=<0|A comp |0> (35)

[0191] <00|A0A1|11>=<0|A comp |1> (36)

[0192] <11|A0A1|00>=<1|A comp |0> (37)

[0193] <11|A0A1|11>=<1|Acomp |1> (38)

[0194] For all possible A0 and A1, Table II below lists the corresponding A0 and A1. comp The transformation table. Since many empty matrices appear in Table II, the measurement overhead can be reduced by combining the reduced set of the operator to be measured (including three single-qubit Pauli matrices and one identity matrix).

[0195] Another aspect of this disclosure includes the distribution of additional CNOTs when using a GC with QSR. In the GC, multiple groups of Pauli strings are created, these groups being extracted from the original groups of strings required for VQE simulation, and can be measured simultaneously. Each group may include a different number of additional CNOTs. Each group may require a different number of measurements, which may be less than the number of Pauli strings within the group. Figure 10 The figures 1000, 1010, 1020, and 1030 illustrate the distribution of the number of additional CNOTs in the number of measurements performed using the GC method for water molecules. The first figure, 1000, illustrates an HF with one additional CNOT. The second figure, 1010, illustrates an HF with nine additional CNOTs. The third figure, 1020, illustrates an HF with nineteen additional CNOTs. The fourth figure, 1030, illustrates an HF with twenty-eight additional CNOTs. The number of additional CNOTs and the number of measurements performed may increase with a larger ansatz size.

[0196] Figure 11 This is a block diagram illustrating an example of a QIP system 1100 according to aspects of this disclosure. The QIP system 1100 may also be referred to as a quantum computing system, a computer device, a trapped ion system, etc.

[0197] QIP system 1100 may include source 1160 that supplies an atomic species (e.g., a plume or flux of neutral atoms) to chamber 1150 having ion trap 1170, which traps ionized (e.g., photoionized) atomic species. In one example, power controller 1140 may provide electrical energy used by source 1160 to generate atomic flux.

[0198] Imaging system 1130 may include a high-resolution imager (e.g., a CCD camera) for monitoring atomic ions as they are provided to the ion trap or after they have been provided to ion trap 1170. In one aspect, imaging system 1130 may be implemented separately from optical controller 1120; however, using image processing algorithms to detect, identify, and label atomic ions using fluorescence may require coordination with optical controller 1120.

[0199] The QIP system 1100 may also include an algorithm component 1110, which can operate in conjunction with other parts of the QIP system 1100 (not shown) to perform quantum algorithms or quantum operations, including stacks or combinations of single-qubit operations and / or multi-qubit operations (e.g., two-qubit operations) and extended quantum computing. Therefore, the algorithm component 1110 can provide instructions to various components of the QIP system 1100 (e.g., to the optical controller 1120) to implement quantum algorithms or quantum operations.

[0200] Now for reference Figure 12 The illustration depicts an example computer device 1200 according to aspects of this disclosure. For example, computer device 1200 may represent a single computing device, multiple computing devices, or a distributed computing system. Computer device 1200 may be configured as a quantum computer (e.g., a quantum information processing (QIP) system), a classical computer, or a combination of quantum and classical computing capabilities. For example, computer device 1200 may be used to process information using quantum algorithms based on trapped ion technology, and thus may implement methods for operating, optimizing, compiling, and / or executing quantum circuits as described above. Figure 11 The example shown illustrates a general example of a computer device 1200 that can implement the various technologies described herein as a QIP system.

[0201] In one example, computer device 1200 may include processor 1210 for performing processing functions associated with one or more features described herein. Processor 1210 may include a single or multiple sets of processors or a multi-core processor. Furthermore, processor 1210 may be implemented as an integrated processing system and / or a distributed processing system. Processor 1210 may include a central processing unit (CPU), a quantum processing unit (QPU), a graphics processing unit (GPU), or a combination of these types of processors. In one aspect, processor 1210 may refer to a general-purpose processor of computer device 1200, which may also include additional processors 1210 to perform more specific functions, such as functions for individual beam control.

[0202] In one example, computer device 1200 may include memory 1220 for storing instructions executable by processor 1210 to perform the functions described herein. In one embodiment, for example, memory 1220 may correspond to a computer-readable storage medium that stores code or instructions to perform one or more of the functions or operations described herein. In one aspect of this disclosure, memory 1220 may be a non-transitory computer-readable medium that stores code or instructions to perform one or more of the functions, techniques, and / or operations described herein. In one example, memory 1220 may include instructions for performing aspects of the methods according to this disclosure. Like processor 1210, memory 1220 may refer to general-purpose memory of computer device 1200, which may also include additional memory 1220 for storing instructions and / or data for more specific functions, such as instructions and / or data for individual beam control.

[0203] Furthermore, computer device 1200 may include communication component 1230 for establishing and maintaining communication with one or more parties using the hardware, software, and services described herein. Communication component 1230 may carry communication between components on computer device 1200, as well as communication between computer device 1200 and external devices, such as devices located on a communication network and / or devices serially or locally connected to computer device 1200. For example, communication component 1230 may include one or more buses, and may also include transmit chain components and receive chain components, respectively associated with a transmitter and a receiver, operable for interfacing with external devices.

[0204] Additionally, computer device 1200 may include data storage 1240, which may be any suitable combination of hardware and / or software for mass storage of information, databases, and programs used in conjunction with the embodiments described herein. For example, data storage 1240 may be a database of an operating system 1260 (e.g., a classic OS or a quantum OS). In one embodiment, data storage 1240 may include memory 1220.

[0205] Computer device 1200 may also include user interface component 1250, operable to receive input from a user of computer device 1200 and further operable to generate output to be presented to the user or provided to different systems (directly or indirectly). User interface component 1250 may include one or more input devices, including but not limited to keyboards, numeric keypads, mice, touch-sensitive displays, digitizers, navigation keys, function keys, microphones, voice recognition components, any other mechanism capable of receiving input from the user, or any combination thereof. Furthermore, user interface component 1250 may include one or more output devices, including but not limited to displays, speakers, haptic feedback mechanisms, printers, any other mechanism capable of presenting output to the user, or any combination thereof.

[0206] In one implementation, the user interface component 1250 can send and / or receive messages corresponding to operations of the operating system 1260. Furthermore, the processor 1210 can execute the operating system 1260 and / or application programs or programs, and the memory 1220 or data storage 1240 can store them.

[0207] When the computer device 1200 is implemented as part of a cloud-based infrastructure solution, the user interface component 1250 can be used to allow users of the cloud-based infrastructure solution to interact remotely with the computer device 1200.

[0208] In some implementations, computer device 1200 may be implemented to perform classical computing, quantum computing, or both. Computer device 1200 may be implemented as a classical computer, a quantum computer, or a hybrid classical-quantum computer.

[0209] In one aspect of this disclosure, the above-described techniques can be executed by a quantum computer, a classical computer, or a hybrid classical-quantum computer. For example, techniques for simulating quantum circuits, optimizing quantum circuits, compiling quantum circuits, and / or executing quantum circuits can be implemented by computer device 1200.

[0210] In some aspects, computer device 1200 and / or one or more sub-components of computer device 1200 may be configured and / or defined as means for implementing the above-described techniques.

[0211] Figure 13Figure 1300 shows the number of measurements for water molecules using the various optimization schemes described above. The number of measurements using the GC partitioning method and QSR optimization is more than two orders of magnitude smaller than the number using QSR optimization alone, while the number using the QWC partitioning method and QSR optimization is approximately one order of magnitude smaller. Figure 1350 shows the additional cost in terms of the number of two-qubit gates associated with the GC partitioning method and QSR, compared to the total number of two-qubit gates using the JW transform. This ratio may decrease as the number of ansatz terms increases.

[0212] In some implementations, the computation can be performed using the HMP2 framework with UCCSD ansatz. Figure 1300 illustrates D as the UCCSD ansatz operator for three different optimization schemes, for the VQE cycle. α The total number of measurements is a function of the number of terms N.

[0213] In some implementations, Figure 1350 shows the average number n of additional two-qubit gates caused by using GC partitioning and QSR. GC+QSR Figure 1350 also shows the total number n of two-qubit gates used to induce UCCSD ansatz states, implemented using the JW transform. JW The above describes n GC+QSR The distribution. Figure 1350 also shows the distribution for the ratio R = n. GC+QSR / (n JW +n GC+QSR The numbers above are drawn.

[0214] In one implementation, the choice between QWC+QSR and GC+QSR can be made based on a resource cost function. For example, QWC+QSR can be chosen if the quality of the two-qubit gate is greater than the required number of measurements divided by the total runtime. Otherwise, GC+QSR can be chosen.

[0215] This disclosure includes a method comprising the steps of: predicting a first set of ansatz terms and a first initial amplitude associated with the first set of ansatz terms; minimizing the energy of the system based on the first set of ansatz terms and the first initial amplitude; calculating the energy or at least one perturbation correction of the wavefunction based on one or more ansatz wavefunctions; determining whether the energy of the system converges; and, in response to determining that the energy of the system does not converge, predicting a second set of ansatz terms and a second initial amplitude associated with the second set of ansatz terms.

[0216] One aspect of this disclosure includes the method described above, and also includes minimizing the energy of the system based on a second set of ansatz terms.

[0217] This disclosure includes any of the methods described above, and also includes generating a simulation output in response to determining the energy convergence of the system.

[0218] This disclosure includes any of the methods described above, wherein predicting the first set of ansatz terms and the first initial magnitude includes using a second-order... -Plesset (MP2) perturbation theory is used for prediction.

[0219] This disclosure includes any of the methods described above, wherein the first set of ansatz comprises a unitarily coupled cluster of ansatz having a single excitation or a double excitation.

[0220] This disclosure includes any of the methods described above, wherein minimizing the energy of the system includes calculating the energy using the variational quantum eigenvalue solver (VQE) method.

[0221] This disclosure includes any of the methods described above, and also includes methods using a hybrid second-order method. The Plesset perturbation (HMP2) method is used to calculate the energy correction operator.

[0222] The aspects of this disclosure include any of the methods described above, as well as compiling the circuit associated with the simulation, optimizing the circuit associated with the simulation, and executing the circuit via a quantum computer.

[0223] This disclosure includes any of the methods described above, wherein predicting the second set of ansatz terms and the second initial magnitude includes increasing the ansatz size of the second set of ansatz terms.

[0224] This disclosure includes any of the methods described above, wherein predicting the second set of ansatz terms and the second initial amplitude includes determining one or more additional ansatz terms to add to the first set of ansatz terms for generating the second set of ansatz terms, and adding the one or more additional ansatz terms to the first set of ansatz terms to generate the second set of ansatz terms, and further includes minimizing the energy of the system based on the second set of ansatz terms and the second initial amplitude.

[0225] The foregoing description of this disclosure is provided to enable those skilled in the art to make or use this disclosure. Various modifications to this disclosure will be apparent to those skilled in the art without departing from its spirit or scope, and the common principles defined herein can be applied to other variations. Furthermore, although elements of the described aspects may be described or claimed in the singular, plural forms are contemplated unless explicitly stated otherwise. Moreover, unless otherwise stated, all or part of any aspect may be used in conjunction with all or part of any other aspect. Therefore, this disclosure is not limited to the examples and designs described herein, but should be given the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for performing a simulation of a chemical system, comprising: The classical processor predicts the first set of ansatz terms and the first initial magnitude associated with the first set of ansatz terms. The classical processor minimizes the energy of the chemical system based on the first set of ansatz terms and the first initial amplitude. One or more perturbation corrections are iteratively applied by the classical processor until the energy converges, including: Calculate the energy or at least one perturbation correction in the wavefunction based on one or more ansatz wavefunctions; Based on at least one calculated perturbation correction, determine whether the energy of the chemical system converges; When the energy of the chemical system does not converge based on the determination, predict the second set of ansatz terms and the second initial magnitude associated with the second set of ansatz terms; Whether the energy of the chemical system converges is determined based on an additional perturbation correction, which is selected based on the second set of ansatz terms and the magnitude of the second initial amplitude; When the energy of the chemical system converges based on the aforementioned additional perturbation correction, the classical processor generates an estimate of the ground-state energy of the particles in the chemical system. The classical processor compiles the circuit based on a simulation of the chemical system, the simulation being based on the ground-state energy estimate of the particles, wherein the compilation includes mapping ansatz terms that lead to convergence of the chemical system to qubit indices in such a way as: The fermion operators contained in the ansatz term are converted into qubit operations using one or more fermion-to-qubit transformations; Assigning qubit operations executable by a quantum processor to the qubit index, wherein the qubit index indicates the placement and application of gates within the circuit; and The circuit is executed by the quantum processor to determine at least one of the chemical reactions, bonding properties, and energy states of the chemical system.

2. The method of claim 1, further comprising minimizing the energy of the chemical system based on the second set of ansatz terms.

3. The method of claim 1, wherein predicting the first set of ansatz terms and the first initial amplitude comprises predicting using second-order Møller-Plesset (MP2) perturbation theory.

4. The method of claim 1, wherein the first group of ansatz terms comprises a unitarily coupled cluster of ansatz with single or double excitation.

5. The method of claim 1, wherein minimizing the energy of the chemical system comprises calculating the energy using a variational quantum eigenvalue solver (VQE) method.

6. The method of claim 5 further comprises calculating the energy correction operator by means of the hybrid second-order Møllar-Plesset perturbation (HMP2) method.

7. The method according to claim 1, further comprising: The circuit is executed by a quantum computer.

8. The method of claim 1, wherein predicting the second set of ansatz terms and the second initial magnitude includes increasing the ansatz size of the second set of ansatz terms.

9. The method of claim 1, wherein predicting the second set of ansatz terms and the second initial magnitude comprises: Determine one or more additional anasatz items to add to the first group of anasatz items, for generating the second group of anasatz items, and Add the one or more additional anasatz items to the first group of anasatz items to generate the second group of anasatz items; and Minimize the energy of the chemical system based on the second set of ansatz terms and the second initial magnitude.

10. A non-transitory computer-readable medium for performing a simulation of a chemical system, the non-transitory computer-readable medium storing instructions that, when executed by a classical processor, cause the classical processor to: Predict the first set of ansatz terms and the first initial magnitude associated with the first set of ansatz terms; Minimize the energy of the chemical system based on the first set of ansatz terms and the first initial amplitude; Iteratively apply one or more perturbation corrections until the energy converges, including: Calculate the energy or at least one perturbation correction in the wavefunction based on one or more ansatz wavefunctions; Whether the energy of the chemical system converges is determined based on at least one calculated perturbation correction; When, based on the determination, the energy of the chemical system does not converge, predict the second set of ansatz terms and the second initial magnitude associated with the second set of ansatz terms; and Whether the energy of the chemical system converges is determined based on an additional perturbation correction, which is selected based on the second set of ansatz terms and the magnitude of the second initial amplitude; When the energy of the chemical system converges based on the aforementioned additional perturbation correction, an estimate of the ground-state energy of the particles in the chemical system is generated. The circuit is compiled based on a simulation of the chemical system, the simulation being based on an estimate of the ground-state energy of the particles, wherein the compilation includes mapping ansatz terms that lead to convergence of the chemical system to qubit indices in such a way as follows: The fermion operators contained in the ansatz term are converted into qubit operations using one or more fermion-to-qubit transformations; A qubit operation that can be performed by a quantum processor is assigned to the qubit index, wherein the qubit index indicates the placement and application of gates within the circuit, and wherein the circuit is executed by the quantum processor to determine at least one of the chemical reactions, bonding properties, and energy states of the chemical system.

11. The non-transitory computer-readable medium of claim 10, further comprising instructions for minimizing the energy of the chemical system based on the second set of ansatz terms.

12. The non-transitory computer-readable medium of claim 10, wherein the instructions for predicting the first set of ansatz terms and the first initial amplitude include instructions for making predictions using second-order Møller-Plesset (MP2) perturbation theory.

13. The non-transitory computer-readable medium of claim 10, wherein the first set of ansatz comprises a unitarily coupled cluster of ansatz having a single excitation or a dual excitation.

14. The non-transitory computer-readable medium of claim 10, wherein the instructions for minimizing the energy of the chemical system include instructions for calculating the energy using a variational quantum eigenfunction solver (VQE) method.

15. The non-transitory computer-readable medium of claim 14, further comprising instructions for calculating an energy correction operator by means of a hybrid second-order Møllar-Plesset perturbation (HMP2) method.

16. The non-transitory computer-readable medium of claim 10, further comprising: Instructions to the circuit are executed by a quantum computer.

17. The non-transitory computer-readable medium of claim 10, wherein the instructions for predicting the second set of ansatz terms and the second initial amplitude include instructions for increasing the ansatz size of the second set of ansatz terms.

18. The non-transitory computer-readable medium of claim 10, wherein the instructions for predicting the second set of ansatz terms and the second initial amplitude comprise instructions for: One or more additional ansatz items are determined to be added to the first group of ansatz items, for generating the second group of ansatz items; as well as Add the one or more additional ansatz items to the first group of ansatz items to generate the second group of ansatz items; and Minimize the energy of the chemical system based on the second set of ansatz terms and the second initial magnitude.

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