Improving the representation accuracy of quantum simulations without additional quantum resources

Abstract

Methods, systems, and apparatus for simulating physical systems. In one aspect, the method includes the actions of: selecting a first basis function set for simulation, wherein the first basis function set includes an active orbital set and a virtual orbital set; defining an extended operator set for simulation, wherein the extended operators in the extended operator set approximate fermion excitations in an active space spanned by the active orbital set and in a virtual space spanned by the virtual orbital set; performing a plurality of quantum calculations to determine a matrix representation of a Hamiltonian that characterizes the system in a second basis function set, using the determined matrix representation of the Hamiltonian, calculating eigenvalues ​​and eigenvectors of the Hamiltonian; and determining properties of the physical system using the calculated eigenvalues ​​and eigenvectors.

Description

Background Art

[0001] This specification relates to quantum computing.

[0002] Applications of quantum computing include quantum simulation. The simulation of quantum systems has applications in a variety of different fields, from drug synthesis to the design of new catalysts and materials. However, simulating complex quantum systems using classical techniques is untenable due to the exponential scaling of the required resources as a function of the system size N. Quantum computers and quantum simulation techniques offer a potential solution to this task. Summary of the invention

[0003] This specification describes methods and systems for improving the representation accuracy of quantum simulations of quantum systems without requiring additional quantum computing resources, such as additional qubits, increased gate complexity, or increased circuit depth.

[0004] In general, one innovative aspect of the subject matter described in this specification can be implemented in a method for simulating a physical system characterized by an electronic structure Hamiltonian, comprising: selecting a first set of basis functions for simulation, wherein the first set of basis functions includes a set of active orbitals and a set of virtual orbitals; defining a set of extension operators for simulation, wherein the extension operators in the set of extension operators approximate fermion excitations in an active space spanned by the set of active orbitals and in a virtual space spanned by the set of virtual orbitals; performing a plurality of quantum calculations to determine: a matrix representation of the electronic structure Hamiltonian in a second set of basis functions, wherein each basis function in the second set of basis functions includes a corresponding extension operator applied to a wave function prepared in the active space, and an overlap matrix in the second set of basis functions, wherein the overlap matrix is ​​a matrix representation of the electronic structure Hamiltonian in the second set of basis functions. Each element represents a corresponding overlap of two basis functions in the second basis function set; wherein determining a matrix representation or overlap matrix of the electronic structure Hamiltonian includes, for each matrix element: determining whether the matrix element includes operators acting on a virtual space or operators acting only on an active space; in response to determining that the matrix element includes operators acting on a virtual space, performing classical calculations to shrink the matrix element to a matrix element including operators acting only on an active space; measuring operators acting only on the active space to determine matrix element values; using the determined matrix representation and the determined overlap matrix of the electronic structure Hamiltonian in the second basis function set, calculating eigenvalues ​​and eigenvectors of the electronic structure Hamiltonian; and determining properties of the physical system using the calculated eigenvalues ​​and eigenvectors.

[0005] Other implementations of these aspects include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more classical and / or quantum computers may be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system that causes the system to perform the actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that cause the device to perform the actions when the instructions are executed by a data processing device.

[0006] The above and other implementations may each optionally include one or more of the following features, either alone or in combination. In some implementations, measuring an operator acting only on the active space to determine a matrix element value includes: preparing a wave function in the active space; and measuring a Pauli operator corresponding to a qubit transformation of the matrix element.

[0007] In some implementations, the method further includes, before calculating the eigenvalues ​​and eigenvectors of the electronic structure Hamiltonian: calculating the eigenvalues ​​and eigenvectors of the determined overlap matrix; removing eigenvalues ​​in the determined overlap matrix that are equal to zero or less than a predetermined threshold to define an updated overlap matrix; and using the updated overlap matrix to calculate the eigenvalues ​​and eigenvectors of the electronic structure Hamiltonian.

[0008] In some implementations, the first set of basis functions also includes a set of core orbitals.

[0009] In some implementations, defining an extended set of operators for simulation further includes selecting a maximum excitation level for the extended set of operators.

[0010] In some implementations, the activity space includes a subspace of the total activity space that is activated into the virtual space.

[0011] In some implementations, measuring the operator that operates only on the active space to determine the matrix element value includes: approximating the operator that operates only on the active space using only cumulative approximation or integrated variational methods; and measuring the approximated operator that operates only on the active space.

[0012] In some implementations, the electronic structure Hamiltonian characterizes the electronic structure of the semiconductor, and wherein simulating the physical system includes simulating properties of the semiconductor. The properties of the semiconductor may include conductivity or resistance.

[0013] In general, another innovative aspect of the subject matter described in this specification can be implemented in a method for simulating a quantum system characterized by a corresponding Hamiltonian. The method includes: obtaining a simulation output from a quantum simulation of the quantum system, wherein the quantum simulation includes a quantum simulation in an active space; and adjusting the simulation output using a plurality of single-particle rotations in the full space by classical computation to obtain an estimated energy of the quantum system characterized by the Hamiltonian, including solving a nonlinear optimization problem, wherein the nonlinear optimization problem includes: an objective function and one or more constraints, the objective function including an expected value of the energy of the quantum system for a measured 2-RDM, and the one or more constraints specifying a unitary rotation of the Hamiltonian orbit.

[0014] Other implementations of these aspects include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more classical and / or quantum computers may be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system that causes the system to perform the actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that cause the device to perform the actions when the instructions are executed by a data processing device.

[0015] The above and other implementations may each optionally include one or more of the following features, alone or in combination. In some implementations, the simulation output includes an estimated energy of the quantum system.

[0016] In some implementations, the Hamiltonian characterizing the quantum system is an electronic structure Hamiltonian, and wherein the simulation output comprises a 2-RDM of a ground state of the quantum system.

[0017] In some implementations, solving the nonlinear optimization problem further includes parameterizing a unitary operator representing the plurality of single-particle rotations as an exponentially anti-Hermitian matrix.

[0018] In some implementations, solving the nonlinear optimization problem includes parameterizing a unitary operator representing a plurality of single-particle rotations as a product of Givens rotations.

[0019] In some implementations, solving the nonlinear optimization problem includes implementing a multi-configuration self-consistent field method.

[0020] In some implementations, one or more constraints produce a normalized physical wave function.

[0021] In some implementations, the method further includes iterating until the energy change between iterations is below a predetermined threshold: providing the adjusted simulation output as input for a subsequent quantum simulation in an active space of the quantum system; and obtaining a subsequent simulation output from the subsequent quantum simulation of the quantum system.

[0022] The subject matter described in this specification can be implemented in a specific way to realize one or more of the following advantages.

[0023] The application of quantum computing in quantum chemistry is usually focused on solving difficult problems in a simplified space called the active space. Restricting to the active space makes it possible to reduce the number of qubits required to solve parts of difficult chemical problems (such as the basic parts). However, this reduction is not accurate. Although the essential physics of the problem can be captured, important effects are ignored, such as electronic spikes that make quantitative accuracy difficult to achieve. Therefore, the calculation and / or simulation results may be inaccurate.

[0024] To capture these additional important effects, additional basis functions are required - which translate into additional qubits and increased quantum gate complexity during the physical realization of the quantum simulation. In most applications, the number of additional basis functions / qubits can be very large. Therefore, determining the exact solution to this type of problem can be extremely difficult and costly - especially when using short- or medium-term quantum computers, for example, noisy intermediate-scale quantum (NISQ) devices where the number of qubits is between tens and hundreds.

[0025] This specification introduces quantum simulation techniques that extend beyond active space without the need for additional qubits or quantum gate complexity. As a result, quantum simulations performed using the presently described techniques reintroduce important physical effects and can achieve improved simulation accuracy. These methods have an exponential advantage over classical methods when dealing with active space and active space references. In addition, optional approximations that can be implemented to improve the efficiency of quantum simulation schemes are introduced. These optional approximations can reduce the number of measurement operations required for quantum simulation schemes, thereby reducing the amount of quantum computing resources used in quantum simulations.

[0026] The techniques described so far are particularly suitable for quantitatively accurate computations of chemical systems on short- or medium-term quantum computers. In the long term, these techniques may also be used to simulate the competitiveness of fault-tolerant methods for quantum systems (e.g., chemical systems). The method is efficiently optimized to run on a hybrid quantum-classical computing device by using a quantum computing device to determine matrix elements and overlap matrices in the active space and using a classical computing device to perform pre-computation and / or post-processing to determine the physical properties of the system based on the matrix elements and overlap matrices.

[0027] The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from the specification, drawings, and claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 An example system for simulating a physical system is depicted.

[0029] Figure 2 is a flow chart of an example process for simulating a physical system using quantum subspace extensions.

[0030] Figure 3 Schematic diagram of the extension of quantum subspace for quantum simulation of quantum systems.

[0031] Figure 4 is a flow chart of an example process for simulating a physical system using full-space orbital relaxation.

[0032] Like reference numbers and designations in the various drawings represent like elements. DETAILED DESCRIPTION

[0033] Overview

[0034] This specification describes techniques for simulating quantum systems, such as systems characterized by electronic structure Hamiltonians. When simulating quantum systems, a discretization of space (or basis set) can be selected to represent the quantum system. The blocks used to divide the space are called basis functions, and many canonical choices of bases are known, such as linear combinations of atomic orbitals and plane waves. Once a basis is selected, the Hamiltonian that characterizes the quantum system can be represented by the selected basis. For example, the electronic structure Hamiltonian can be written in its canonical form as:

[0035]

[0036] where each index i, j, k, l corresponds to a basis function in the selected basis, h ij and h ijkl represents the standard integral over the basis functions involved, and the ladder operator a satisfies the canonical fermion anti-exchange relation

[0037] The accuracy that can be achieved in the simulation of a quantum system depends on the number of basis functions in the chosen basis. That is, in general, increasing the number of basis functions will increase the accuracy of the simulation. However, using too many basis functions may make the simulation task impractical or may waste resources when using more intensive processing methods.

[0038] One approach first used in classical quantum chemistry and adopted by the quantum computing community is the active space approximation. The physical intuition behind the active space approximation is that the discretized space can be divided into a part that exhibits strong correlations or entanglement - the active space - and a part that is important but exhibits low-rank contributions that can be well perturbed - the virtual space. However, on classical computers, the size of the elementary quantum components or active space is still finite, and so far, there are no virtual contributions on quantum computers.

[0039] This specification describes quantum simulation techniques and systems that go beyond active space approximation without additional qubits or gate complexity. One example technique includes using quantum subspace expansion to systematically reintroduce contributions from virtual space to improve simulation accuracy. Another example technique uses orbital relaxation to remove active space approximation and reduce circuit depth.

[0040] Example Hardware

[0041] Figure 1 An example system 100 for simulating quantum systems is depicted. Example system 100 is an example of a system implemented as classical and quantum computer programs on one or more classical computers and quantum computing devices at one or more locations in which the systems, components, and techniques described in this specification may be implemented.

[0042] System 100 may receive input data specifying a physical system to be modeled or simulated, such as input data 106. For example, the received data may represent a material, such as a metal or polymer, a single molecule, or a chemical. The input data may include data representing a Hamiltonian that characterizes the physical system to be modeled or simulated, such as an electronic structure Hamiltonian.

[0043] The system 100 may generate output data representing the results of the simulation of the physical system of interest, e.g., output data 108. The generated output data may be provided for further processing or analysis. For example, where the physical system is a material (e.g., a metal or a polymer), the generated output data may include data representing a simulated ground state of the physical system, which may be used to determine properties of the material, e.g., its resistance or conductivity. As another example, where the physical system is a chemical, the generated output data may be used to determine properties of the chemical, e.g., a chemical reaction rate.

[0044] The system 100 includes quantum hardware 102 in data communication with a classical processor 104. For convenience, the classical processor 104 and the quantum computing hardware 102 are shown as separate entities, however in some implementations, the classical processor 104 may be included in the quantum computing hardware 102, e.g., the quantum computing hardware 102 may include one or more components for performing classical computing operations.

[0045] Quantum hardware 102 includes components for performing quantum computations, such as the quantum simulation processes described herein. For example, quantum hardware 102 includes a plurality of qubits 110 and a control device 112 for controlling qubits 110 and causing algorithmic operations or quantum computations to be performed.

[0046] The plurality of qubits 110 are physical qubits used to perform algorithmic operations or quantum computations. The specific physical implementation of the qubits 110 included in the quantum computing hardware 102 and how they interact with each other depends on various factors, including the type of quantum computation being performed by the quantum computing hardware 102 (which in turn depends on the physical system being simulated). For example, in some implementations, qubits may include qubits physically implemented by atoms, molecules, or solid-state quantum systems. In other implementations, qubits may include superconducting qubits, such as Gmon qubits or semiconductor qubits. In other implementations, ion traps, photonic devices, or superconducting cavities (with which states can be prepared without the need for qubits) may be used. Further examples of physical implementations of qubits include fluxon qubits, silicon quantum dots, or phosphorus impurity qubits.

[0047] The type of control device 112 included in quantum hardware 102 depends on the type of qubit 110 included in quantum hardware 102. For example, in some cases, qubit 110 can be frequency tunable. In these cases, each qubit can have an associated operating frequency that can be adjusted using one or more control devices 112, such as an excitation pulse generator and a control line that couples the qubit to the excitation pulse generator. Example operating frequencies include qubit idle frequency, qubit interaction frequency, and qubit readout frequency. Different frequencies correspond to different operations that a qubit can perform. For example, setting the operating frequency to a corresponding idle frequency can cause the qubit to enter a state in which it does not strongly interact with other qubits and can be used to perform a single qubit gate. As another example, in the case where qubits interact through a coupler with a fixed coupling, the qubits can be configured to interact by setting their corresponding operating frequencies to a certain gate-dependent frequency that is detuned from their common interaction frequency. In other cases, for example, when qubits interact through tunable couplers, the qubits can be configured to interact with each other by setting the parameters of their respective couplers to enable interaction between the qubits, and then by setting the respective operating frequencies of the qubits to some gate-dependent frequency that is detuned from their common interaction frequency. Such interaction can be performed to perform a multi-qubit gate.

[0048] Control device 112 may also include devices such as readout resonators that are configured to perform measurements on qubits 110 and provide the measurements to classical processor 104 for processing and analysis.

[0049] The classical processor 204 includes components for performing classical computations, such as the classical post-processing procedures described in this specification.

[0050] Programming Hardware: Example Processing Using Quantum Subspace Extensions to Simulate Quantum Systems

[0051] Figure 2is a flow chart of an example process 200 for simulating a quantum system characterized by a corresponding Hamiltonian using quantum subspace expansion. For convenience, process 200 is described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to the present specification, such as Figure 1 The system 100 may perform the process 200 .

[0052] The system selects a first set of basis functions for simulation (step 202). The system can then define a Hamiltonian using the selected first set of basis functions. For example, as described above, in some cases, the Hamiltonian can be an electronic structure Hamiltonian. In these cases, the electronic structure Hamiltonian can be written in the canonical form given in equation (1) above, where each index value of i, j, k, l in equation (1) corresponds to a corresponding basis function.

[0053] The first set of basis functions includes the core orbitals Activity Track and virtual tracks In typical quantum chemical calculations on quantum computers (i.e., calculations different from those described in this specification), the core orbitals are assumed to be doubly occupied and their contributions are integrated into the effective fields felt in both the active and virtual spaces. In these typical quantum chemical calculations, the virtual orbitals are neglected, and the problem is precisely in the active space. Solved within.

[0054] The system can determine which elements of the first basis function set belong to each subset by performing a classical pre-calculation or For example, the system can perform mean field calculations, such as Hartree Fock, on a quantum system of interest. Performing such calculations produces a new set of orbitals constructed from the original orbitals that are well ordered in terms of energy. The orbitals with the lowest energy are less likely to participate in bonding and are labeled or designated as core orbitals. The orbitals that are closest in energy to the highest occupied orbital can be labeled or designated as active orbitals in the active space. The remaining orbitals are labeled or designated as virtual orbitals. In some cases, this classical pre-calculation can be enhanced by going beyond the mean field level, rather than sorting by occupancy in the natural orbitals. In addition, the classical pre-calculation can be further refined by approximately accounting for orbital entanglement using the method of the density matrix renormalization group.

[0055] System defined extension operator O i (step 204). The expansion operator includes the active space (the space spanned by the active set of tracks) and virtual space The operator that approximates the fermion excitation on the space spanned by the set of virtual orbitals. For example, the extended operator set can be given by the following equation (2).

[0056]

[0057] In equation (2), a denotes the fermion annihilation and creation operator. The number of expansion operators in the defined set is determined by the number of primitive basis functions, the combinations of which have been marked as core, active or virtual by the classical pre-step described above. and Counting the indices in equation (2) by the size of the set, the number of expansion operators in the defined set can be precisely determined based on these set sizes and the fact that items with 2 indices and 4 indices are selectively extracted from the set.

[0058] The system performs a plurality of quantum computations to determine a matrix representation of a target operator (corresponding to an observable target for simulation) in a second set of basis functions (step 206). For convenience, the following description continues to use the example of the electronic structure Hamiltonian as the target operator (where energy is the observable target), however other target operators and observable values ​​may be used (depending on the simulation being performed). Computing such matrix elements by classical computing devices is typically computationally expensive, particularly for states that exhibit strong correlations and / or entanglement. In contrast, using quantum computations to determine matrix elements inherently accounts for strong correlations and / or entanglement.

[0059] By replacing the corresponding expansion operator O i Applied to the reference wave function |Ψ ref > to define each basis function in the second basis function set, which can be in the active space Internally (effectively) prepare the reference wave function |Ψ ref >. That is, the second basis function set can be composed of {O i |Ψ ref The matrix representation H of the electronic structure Hamiltonian is determined by the measurement operation and has matrix elements given by the following equation (3).

[0060]

[0061] To determine each matrix element H ij , the system can repeatedly prepare the wave function |Ψ ref> (by preparing qubits in the quantum computing device in the corresponding quantum state) and measuring the transformed Pauli operators corresponding to the matrix elements by the Jordan-Wigner transformation or an equivalent transformation. In some implementations, more general measurement techniques may be used, such as phase estimation or ancilla assisted estimation.

[0062] In some implementations, the second basis function set {O i |Ψ ref .} may be non-orthogonal. In these implementations, the system may further determine an overlap (or metric) matrix S in the second basis function set (to ensure that the problem is well defined) (step 208). Each element of the overlap matrix S represents the corresponding overlap of two basis functions in the basis function set. Each element may be formed by a measurement operation and may be given by the following equation (4).

[0063]

[0064] Again, to determine each matrix element S ij , the system can repeatedly prepare the wave function |Ψ ref . (by preparing the qubits in the quantum computing device in the corresponding quantum state) and measuring the Pauli operator corresponding to the transformation of the matrix elements by the Jordan-Wigner transformation or an equivalent transformation.

[0065] The expansion operator defined in step 204 and equation (2) above includes the following: and virtual space However, by definition, the wave function |Ψ ref >No effect on virtual space Therefore, the expansion operator acting on the virtual space can be contracted using classical precomputation, for example, by applying Wick's theorem or equivalent to the Hamiltonian, so that each matrix element H ij and S ij Quantum computations can only be performed in active space.

[0066] For example, in some implementations, step 206 may include determining by measurement the matrix elements given by equation (5) below.

[0067]

[0068] In equation (5), the Greek letters represent virtual orbitals. In this case, performing a classical precomputation may include contracting the operator in equation (5) on the virtual space using Wick's theorem to obtain an alternative representation as given in equation (6).

[0069]

[0070] In equation (6), Δ ξη,μv =δ ξv δ ημ -δ ξμ δ ηv , means that x(y) changes to another value in the sum (for example, For x), ,… denotes the active space with the wave function |Ψ ref >Related expected values. represents the 4-electron reduced density matrix (4-RDM), represents the 3-electron density matrix (3-RDM), and Represents the 2-electron density matrix (2-RDM).

[0071] The expansion operator defined in equation (2) includes both inside and outside the active space with respect to the reference wave function |Ψ ref > single and double excitations, as shown in equation (2). However, in some implementations, the number of excitations can vary to any level within or outside the active space or within a subdivision thereof. That is, the system can select a maximum excitation level k for any k-th extended operator set. For k-th level of excitation, the number of desired measurements included in process 200 will grow to As this number increases, the expected accuracy of both the ground state and the approximation to the excited states will increase, but the cost of the measurement will also increase.

[0072] The system calculates eigenvalues ​​and eigenvectors of the electronic structure Hamiltonian using the determined matrix representation H of the electronic structure Hamiltonian and the determined overlap matrix S (step 210). That is, the system can perform classical calculations to solve the generalized eigenvalue problem given by equation (7) below.

[0073] HC=SCE (7)

[0074] In equation (7), C represents the base {O i |Ψ ref>}, and E denotes the diagonal matrix of eigenvalues. In some implementations, a preprocessing step, such as regularized diagonalization, may be performed to remove zero or near-zero eigenvalues ​​of the matrix S and generate an updated overlap matrix S' before solving the generalized eigenvalue problem in the well-conditioned subspace.

[0075] The system uses the calculated eigenvalues ​​and eigenvectors to determine the characteristics of the quantum system (step 212). For example, in some cases, the electronic structure Hamiltonian can characterize the electronic structure of a semiconductor. In these cases, simulating the physical system can include simulating the properties of the semiconductor, such as simulating the conductivity or resistance of the semiconductor. Such simulation results can be used to manufacture semiconductor devices, such as integrated circuits. Another example is that in some cases, the electronic structure Hamiltonian can characterize a catalyst. In these cases, simulating the physical system can include simulating the properties of the catalyst, such as simulating the catalytic activity. Such simulation results can be used to prepare catalysts, such as electrocatalysts or biocatalysts.

[0076] Figure 3 is a schematic diagram 300 of the presently described virtual quantum subspace extension technique for quantum simulation of quantum systems, eg, steps 202 - 210 of the example process 200 described above.

[0077] As described above with reference to step 202 of the example process, the technique separates the track of a quantum system (eg, including track 102) into its core component 104a, active component 104b, and virtual component 104c.

[0078] As described above with reference to steps 204-208 of example process 200, quantum computing device 106 simulates a quantum system in active space (i.e., the state of the quantum system represented by a wave function). Additional measurements are made in this space. Classical computer 108 combines these additional measurements with classical post-processing of the data from virtual space 104c to improve the simulation without using additional qubits or circuit depth.

[0079] As described above with reference to steps 210 - 212 of example process 200 , the resulting improved simulated wave function(s) may be stored in a hybrid quantum classical representation and may be used to derive target properties of the wave function(s).

[0080] Programming Hardware: Cumulants and Restricted Activity Space Approximation

[0081] In some implementations, one or more approximations may be introduced into the example process 200 described above to improve traceability.

[0082] An example approximation involves converting the activity space Divided into being stimulated into virtual space The first part and the second part of the activity space of the core orbital which can be considered as related That is, in some implementations, the system can Execute under the condition Figure 2 Example process 200. Partitioning the activity space in this way reduces the scaling of the number of measurements required to determine the matrices H and S to the original cost of computing the eigenvalues ​​plus the first part The size of For small size Can significantly reduce costs.

[0083] Another approximation includes estimating the matrix elements of the reduced density matrix (RDM) by cumulative approximation or integrated variational methods. For example, a series of approximations to a 4-RDM, such as the 4-RDM defined in equation (6), can be formed using the product of a lower RDM and a perturbation correction. Such truncation can reduce the number of active space orbits returned to the measurement. The number of terms required is small, but some approximation to the target observable (eg, energy) is introduced. Another alternative is to randomly sample elements of the approximation of the 4-RDM for measurement, thereby increasing accuracy over computation time.

[0084] Programming Hardware: Example Processing for Simulating Quantum Systems Using Full-Space Orbital Relaxation

[0085] Figure 4 is a flow chart of an example process 400 for simulating a quantum system characterized by a corresponding Hamiltonian using full-space orbital relaxation. For convenience, process 400 is described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to the present specification, such as Figure 1 The system 100 may perform the process 400 .

[0086] The system obtains a simulation output from a quantum simulation of a quantum system (step 402). The quantum simulation can be a quantum simulation in an active space, for example, a quantum simulation performed using an active space approximation. The simulation output can be an estimated energy of the quantum system, for example, an estimated energy associated with a Hamiltonian in the active space. For example, in some implementations, the Hamiltonian characterizing the quantum system can be an electronic structure Hamiltonian, as given by equation (1) above. In these implementations, the simulation output can include a 2-RDM (Hamiltonian obtained by integrating virtual degrees of freedom and core degrees of freedom) from the ground state of the corresponding active space Hamiltonian. The system can obtain the simulation output from active space methods such as fully active space configuration interaction (CAS-CI), multi-configuration self-consistent field method (MCSCF), density matrix renormalization group (DMRG), quantum Monte Carlo (QMC), or more generally, from any method that outputs a 2-RDM (or N-representable 2-RDM).

[0087] The system performs classical computations to adjust the simulation output using multiple single-particle rotations U in full space to obtain an estimated energy of the quantum system characterized by the corresponding Hamiltonian (i.e., the Hamiltonian in full space) (step 404). Due to Thouless's theorem, multiple single-particle rotations U can be effectively implemented as rotations of the basis of the Hamiltonian (in full space), for example, The rotation of this basis can be formulated as a nonlinear optimization problem. The nonlinear optimization problem follows the variational principles of quantum mechanics by minimizing an objective function based on the energy expectation value of the measured 2-RDM with respect to a number of constraints (unitary rotation of the orbital) that ensure that the wave function preserves the normalized, physical, and integral fermion rotation. These constraints also ensure that the equations are equivalent to classical evaluations once the 2-RDM is known. The nonlinear optimization problem can be given by the following equation (8).

[0088]

[0089] Submission

[0090]

[0091] In equation (8), U represents the unitary transformation of the rotation basis, i.e., multiple single-particle rotations, u i,i′ is the representation of U in the basis (e.g., u i,i′ is the i-i'th item of the matrix representation of U), h ij and h ijkl As defined in equation (1) above, and and 2-RDM and 1-RDM representing the ground state active space wave function (which can be obtained at step 402). In some implementations, the system can solve the nonlinear optimization problem by implementing a second-order approximation of u, for example,

[0092] In some implementations, the system can solve the nonlinear optimization problem given by equation (8) by parameterizing the unitary transformation U as an exponential anti-Hermitian matrix. For example, the system can set

[0093] U=e X

[0094]

[0095]

[0096] In equation (9), X represents the generator of the unitary function U and is an anti-Hermitian matrix.

[0097] In other implementations, the system can parameterize the unitary matrix U using Givens rotations. This parameterization uses a set of angles {θ} associated with a set of non-redundant orbital rotation generators. For a 2-RDM obtained from an exact diagonalization of the active space Hamiltonian, the only non-redundant parameters are the single-particle generators associated with the orbital pairs involving rotations from active space to virtual space and active space to core space. Therefore, the unitary matrix U in equation (8) can be expressed as a product of Givens matrices:

[0098]

[0099] The optimal rotation of a single angle relative to the input 2-RDM, 1-electron integral and 2-electron integral and the scanning procedure to find the energy minimization U can be calculated using known techniques, for example, based on a full optimization of successive Jacobian rotations using the multi-configuration self-consistent field method (MCSF) method for ground and excited states.

[0100] In some implementations, the system can iterate between solving the active-space Schrödinger equation (step 402) and the full-space single-particle rotation (404), for example, iteratively performing the example process 400. When the system iteratively performs the example process 400, at the end of each iteration, a new basis determined by minimizing the expectation value of the unitary rotation of the Hamiltonian with respect to the orbit (determined by solving Equation 8 above) is used as an input to solving the active-space Schrödinger equation in a subsequent iteration. The iterative process can stop when the energy change between iterations is below a predetermined threshold. In other implementations, the system can perform a single orbital relaxation process (step 404) once as a post-processing step.

[0101] Due to the variational principle, it is guaranteed that the orbitals are relaxed in a single step in post-processing to reduce energy. In addition, if the ground state of the active space is not reached due to the approximate wave function ansatz, the system can include additional single-particle orbital rotations between orbital pairs in the active space in the relaxation. Including these rotations will correspond to additional linear depth circuits, which will be perfectly executed on the quantum computer in step 402. However, any circuit that includes single-particle rotations at the end of the circuit can be shortened by replacing the single-particle rotations at the end of the circuit with a classical post-processing step based on the above-mentioned single-particle rotations (e.g., in the active space instead of the full space).

[0102] In some implementations, example process 400 can be combined with one or more techniques described above with reference to example process 200. For example, step 402 can be performed using the techniques described in steps 202-208 of example process 200, where, in example process 200 (as described in steps 206 and 208), the contraction of the expansion operator acting on the virtual space by classical pre-computation can be replaced by orbital relaxation by classical post-processing (step 404). Then, the following steps can be performed: Figure 3 Steps 210 and 212.

[0103] The digital and / or quantum subject matter and implementation of digital functional operations and quantum operations described in this specification may be implemented in digital electronic circuits, suitable quantum circuits or more generally quantum computing systems, in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in a combination of one or more structural equivalents or in a combination of one or more of them. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptographic system or a quantum simulator.

[0104] Implementations of the digital and / or quantum subject matter described in this specification may be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution, or to control the operation of a data processing apparatus. The digital and / or quantum computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or in addition, the program instructions may be encoded on an artificially generated propagated signal capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, generated to encode digital and / or quantum information for transmission to an appropriate receiver device for execution by the data processing apparatus.

[0105] The terms quantum information and quantum data refer to information or data carried, held or stored by a quantum system, wherein the smallest non-trivial system is a qubit, i.e., a system that defines a unit of quantum information. It should be understood that the term "qubit" includes all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., having two or more levels. For example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational ground state is identified by the ground state and the first excited state, however, it will be appreciated that in other settings, it is possible for the computational state to be identified by a higher level excited state.

[0106] The term "data processing device" refers to digital and / or quantum data processing hardware, including various devices, apparatuses and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers and combinations thereof. The device may also be or further include a dedicated logic circuit, such as an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit) or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a specific quantum system. In particular, a quantum simulator is a special-purpose quantum computer that does not have the ability to perform general-purpose quantum computations. In addition to the hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, for example, code constituting processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more thereof.

[0107] A digital computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages, declarative or procedural languages, and may be deployed in any form, including as a stand-alone program or as a module, component, subroutine or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or may be written in a quantum programming language, such as QCL or Quipper.

[0108] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program may be stored in a file portion that stores other programs or data, for example, one or more scripts stored in a markup language document, in a single file dedicated to the program, or in multiple coordinated files, for example, files storing one or more modules, subroutines, or partial codes. A digital and / or quantum computer program may be deployed to execute on a digital or quantum computer, or on multiple digital and / or quantum computers located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can transmit quantum data using a quantum system (e.g., qubit). Typically, a digital data communication network cannot transmit quantum data, however a quantum data communication network can transmit quantum data and digital data.

[0109] The processes and logic flows described in this specification may be performed by one or more programmable digital and / or quantum computers, and may be operated using one or more digital and / or quantum processors, as appropriate, to operate on input digital and quantum data and generate output, and to execute one or more digital and / or quantum computer programs to perform functions. The processes and logic flows may also be performed by dedicated logic circuits (e.g., FPGAs or ASICs) or quantum simulators, and the apparatus may also be implemented as dedicated logic circuits, or by a combination of dedicated logic circuits or quantum simulators and one or more programmed digital and / or quantum computers.

[0110] For a system consisting of one or more digital and / or quantum computers, “configured to” perform a specific operation or action means that the system has installed thereon software, firmware, hardware, or a combination thereof, which, in operation, causes the system to perform the operation or action. For one or more digital and / or quantum computer programs configured to perform a specific operation or action, it means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer can receive instructions from a digital computer that, when executed by the quantum computing device, cause the device to perform the operation or action.

[0111] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general or special purpose digital and / or quantum processor or both, or any other type of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory, or a quantum system suitable for transmitting quantum data (e.g. photons), or a combination thereof.

[0112] Elements of a digital and / or quantum computer include a central processing unit for executing or performing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory may be supplemented or incorporated by a dedicated logic circuit or a quantum simulator. Typically, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from one or more mass storage devices for storing digital and / or quantum data, or to transfer digital and / or quantum data to one or more mass storage devices, or both, for example, magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer does not require such devices.

[0113] Digital and / or quantum computer readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including, for example, semiconductor memory devices, such as EPROM, EEPROM and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM optical disks; and quantum systems, such as trapped atoms or electrons. It is understood that quantum memory is a device capable of storing quantum data for a long time with high fidelity and high efficiency, for example, light-matter interfaces for transmission and matter for storing and preserving quantum characteristics (such as superposition or quantum coherence) of quantum data.

[0114] Control of the various systems or portions thereof described in this specification may be implemented in a digital and / or quantum computer program product that includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions thereof described in this specification may each be implemented as an apparatus, method, or system that may include one or more digital and / or quantum processing devices and memory to store executable instructions for performing the operations described in this specification.

[0115] Although this specification contains many specific implementation details, these details should not be interpreted as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to a particular implementation. Certain features described in this specification in the context of separate implementations may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented separately in multiple implementations or in any suitable sub-combination. In addition, although the above features may be described as working in certain combinations, or even initially claimed to be so, in some cases, one or more features from the claimed combination may be deleted from the claimed combination, and the claimed combination may be directed to a sub-combination or a variation of a sub-combination.

[0116] Similarly, although operations are described in a particular order in the accompanying drawings, this should not be understood as requiring that such operations be performed in the particular order or sequence shown, or that all illustrated operations be performed to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous. In addition, the separation of various system modules and components in the above implementations should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated into a single software product, or packaged into multiple software products.

[0117] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve the desired results. As an example, the processes described in the accompanying drawings do not necessarily require the particular order or sequence shown to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for quantum simulation of a physical system characterized by a corresponding Hamiltonian, the method comprising: selecting a first basis function set for simulation, wherein the first basis function set includes i) an active orbital set and ii) a virtual orbital set, wherein orbitals closest in energy to a highest occupied orbital are marked as active orbitals in an active space, and remaining orbitals other than core orbitals and active orbitals are marked as virtual orbitals; defining an extended operator set for simulation, wherein extended operators in the extended operator set approximate fermion excitations in an active space spanned by the active orbital set and in a virtual space spanned by the virtual orbital set; performing a plurality of quantum computations to determine a matrix representation of the Hamiltonian in a second set of basis functions, wherein each basis function in the second set of basis functions comprises a corresponding expansion operator applied to a wave function prepared in the active space, and wherein, for each matrix element, determining the matrix representation of the Hamiltonian comprises: Determine whether the matrix elements include operators acting on virtual space or only on active space, In response to determining that the matrix element includes an operator that acts on the virtual space, performing a classical calculation to shrink the matrix element to a matrix element that includes operators that act only on the active space, and Measurements are made on operators acting only on the active space to determine the values ​​of the matrix elements; Using the determined matrix representation of the Hamiltonian in the second set of basis functions, computing eigenvalues ​​and eigenvectors of the Hamiltonian; and Use the computed eigenvalues ​​and eigenvectors to determine the properties of the physical system.

2. The method of claim 1 further comprising performing a plurality of calculations to determine an overlap matrix in the second set of basis functions, wherein each element of the overlap matrix represents a corresponding overlap of two basis functions in the second set of basis functions, wherein: For each matrix element, determining the overlap matrix involves: determining whether the matrix elements include operators acting on virtual space or only on active space; In response to determining that the matrix element includes an operator that operates on the virtual space, performing a classical calculation to shrink the matrix element to a matrix element that includes operators that operate only on the active space; and Measurements are made only on operators that act on the active space to determine the values ​​of the matrix elements.

3. The method according to claim 2, wherein: Using the determined matrix representation of the Hamiltonian in the second set of basis functions, computing the eigenvalues ​​and eigenvectors of the Hamiltonian includes: Using the determined matrix representation of the Hamiltonian in the second set of basis functions and the determined overlap matrix, eigenvalues ​​and eigenvectors of the Hamiltonian are calculated.

4. The method according to claim 1 or claim 2, wherein: Measuring operators that act only on the active space to determine the values ​​of matrix elements includes: preparing wave functions in active space; and Measure the Pauli operator corresponding to the qubit transformation of the matrix elements.

5. The method according to claim 2 further comprises, before calculating the eigenvalues ​​and eigenvectors of the Hamiltonian: calculating the eigenvalues ​​and eigenvectors of the determined overlap matrix; removing eigenvalues ​​in the determined overlap matrix that are equal to zero or less than a predetermined threshold value to define an updated overlap matrix; as well as Use the updated overlap matrix to compute the eigenvalues ​​and eigenvectors of the Hamiltonian.

6. The method according to claim 1, wherein: The first basis function set also includes a core orbital set.

7. The method according to claim 1, wherein: Defining an extended operator set for simulation also includes selecting a maximum excitation level for the extended operator set.

8. The method according to claim 1, wherein: The activity space includes the subspaces in the total activity space that are activated into the virtual space.

9. The method according to claim 1, wherein: Measuring operators that act only on the active space to determine the values ​​of matrix elements includes: Use cumulative approximation or integrated variational methods to approximate operators that act only on the active space; and The measurements act only on approximate operators in the active space.

10. The method according to claim 1, wherein: The Hamiltonian characterizes the electronic structure of the semiconductor, and wherein simulating the physical system includes simulating properties of the semiconductor.

11. The method according to claim 10, wherein: Properties of semiconductors include conductivity or resistance.

12. An apparatus for quantum simulation of a physical system characterized by a corresponding Hamiltonian, the apparatus comprising: Quantum hardware; as well as One or more classical processors; The device is configured to perform operations including the method described in any one of claims 1 to 11.

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