A Stochastic Simulation Method and Device for an Open Quantum System

Abstract

The present application provides a method and apparatus for stochastic simulation of an open quantum system. In the method for stochastic simulation of an open quantum system, first, the total Hamiltonian of the new system is decomposed into a linear combination of Pauli operators, and a combination of quantum gates is constructed using the coefficients of the linear combination. Then, a classical sampling method with Pauli operators as sampling samples is run, and a quantum circuit is constructed using the combination of quantum gates. According to the operation result of the quantum circuit, the mean value of the target observable in the open quantum system is finally obtained. The present application uses a shallower quantum circuit to simulate large-scale practical quantum system dynamics problems, alleviates the limitation of quantum chip resources, uses error mitigation techniques to reduce the circuit depth, allocates part of the calculation process to a classical computer, and only runs the process involving observables on a quantum computer, and combines the two parts of the calculation results on a classical computer, saving quantum hardware resources.

Description

Technical Field

[0001] One or more embodiments of this specification relate to the field of quantum computers, and in particular, to a method and device for stochastically simulating an open quantum system. Background Art

[0002] In existing quantum systems, the calculations of observables are all for corresponding closed quantum systems. In a closed system, the evolution of the quantum state is unitary, periodic, and oscillatory, and can be described by the Schrödinger equation. During the evolution process, the quantum state is always a pure state. However, an open quantum system is inevitably affected by environmental noise, so that the pure state becomes a mixed state. This process is non-unitary, non-periodic, and either oscillatory decay or continuous decay. Generally, under the assumptions of weak coupling with the environment and the environment being Markovian, this process can be described by the Lindblad master equation.

[0003] When the scale of the open quantum system is large, the computational amount of the Lindblad master equation increases exponentially with the system scale. Classical dynamic simulation methods are often limited by accuracy, memory, and simulation speed, and it is difficult to effectively simulate an open quantum system. In this case, quantum algorithms have become a promising alternative that may reduce the exponentially growing cost. However, in the case of requiring high-order accuracy, quantum algorithms may require many auxiliary qubits, complex quantum control logic, or involve additional quantum state amplitude amplification processes. Therefore, compared with the simulation algorithms of closed quantum systems, the implementation of these algorithms is much more complex, making it difficult to implement the simulation of open quantum systems in quantum computers. Summary of the Invention

[0004] This application describes a method and device for stochastically simulating an open quantum system, which can solve the above technical problems.

[0005] According to a first aspect, there is provided a method for stochastically simulating an open quantum system, the method including:

[0006] Using the Lindblad master equation as a bridge with a stochastic differential equation, approximately transforming the open quantum system into a new closed system, and deriving the total Hamiltonian of the new closed system as the newly constructed total Hamiltonian from it, where the newly constructed total Hamiltonian incorporates the system Hamiltonian of the open quantum system and the dissipation term of the interaction between the system and the environment, and the dimension of the new closed system is higher than that of the open quantum system;

[0007] Decomposing the time evolution operator of the newly constructed total Hamiltonian into a linear combination of Pauli operators, and constructing a combination of quantum gates using the coefficients of the linear combination, where the combination of quantum gates is used to calculate the target observable in the open quantum system;

[0008] Run a classical sampling method with Pauli operators as sampling samples to obtain the sampled Pauli operators, map the sampled Pauli operators into a combination of the quantum gates, construct a quantum circuit using the combination of the quantum gates, and obtain the mean value of the target observable in the open quantum system according to the operation result of the quantum circuit.

[0009] In some embodiments, the decomposing the time evolution operator of the newly constructed total Hamiltonian into a linear combination of Pauli operators and constructing a combination of quantum gates using the coefficients in the linear term combination specifically includes:

[0010] Using the Lie-Trotter iteration decomposition and Richardson extrapolation method, decompose the time evolution operator of the newly constructed total Hamiltonian into a linear superposition of a correction operator and the product of the Lie-Trotter formula;

[0011] Construct the correction operator through the identity matrix and the local truncation error;

[0012] Decompose the linear term in the local truncation error into a linear term combination of Pauli operators, and construct the combination of the quantum gates using the coefficients in the linear term combination.

[0013] In some more specific embodiments, the using the Lie-Trotter iteration decomposition and Richardson extrapolation method to decompose the time evolution operator of the newly constructed total Hamiltonian into a linear superposition of a correction operator and the product of the Lie-Trotter formula, and representing the correction operator through the identity matrix and the local truncation error specifically includes:

[0014] According to the derived , combine the leading term rotation method and the Richardson extrapolation method to design a classical sampling algorithm and the corresponding quantum Monte Carlo circuit to obtain the mean value of any observable ;

[0015] The principle of quantum simulation is based on:

[0016]

[0017]

[0018] and is the intermediate term for constructing ;

[0019] where the 2m-order Lie-Trotter-Suzuki product formula is iteratively calculated according to the following formula:

[0020]

[0021] Among them, ,

[0022] ,

[0023] In the above formula, The main term of Its meaning is The lower-order term of , is the remainder The higher-order term of is the identity matrix, is the identity matrix, is a parameter related to the order, is the order, is the correction operator, among which, is The conjugate transpose of is the total Hamiltonian The unitary time-evolution operator at time t, m is an integer.

[0024] In some embodiments, decomposing the linear term in the local phase error into a linear combination of Pauli operators, and constructing a combination of quantum gates using the coefficients in the linear combination specifically includes:

[0025] Decompose the local truncation error into a linear combination of Pauli operators where is the first coefficient in the linear combination; is the first coefficient in the linear combination;

[0026] Calculate the absolute value sum of the first coefficient in the linear combination ;

[0027] Determine the rotation angle through the absolute value sum of all coefficients , ;

[0028] Determine the second coefficient through the first coefficient and the rotation angle , ;

[0029] Construct a combination of quantum gates using the first coefficient , the second coefficient and the Pauli operator , , is the quantum gate corresponding to the Pauli operator , is the first coefficient The sign of determines the rotation direction of the quantum gate.

[0030] In some embodiments, the method includes running a classical sampling method with Pauli operators as sampling samples to obtain the sampled Pauli operators, mapping the sampled Pauli operators into a combination of the quantum gates, constructing a quantum circuit using the combination of the quantum gates, and obtaining the mean value of the target observable in the open quantum system according to the operation result of the quantum circuit, specifically including:

[0031] Running a classical sampling method with Pauli operators as sampling samples on a classical computer to obtain the sampled Pauli operators, mapping the sampled Pauli operators into a combination of the quantum gates to obtain a quantum circuit;

[0032] Executing the quantum circuit on a quantum computer to obtain the expected value of the target observable;

[0033] Repeatedly execute the above operations, and determine the mean value of the target observable in the open quantum system according to the expected values obtained from multiple operation processes.

[0034] In some embodiments, the method of approximately mapping the Lindblad master equation to the newly constructed total Hamiltonian of the open quantum system with a stochastic differential equation as a bridge specifically includes:

[0035] Performing a randomized approximation on the Lindblad master equation to transform it into a stochastic differential equation;

[0036] Based on the stochastic differential equation, derive the newly constructed total Hamiltonian of the open quantum system .

[0037] According to the second aspect, a stochastic simulation device for an open quantum system is provided, and the device includes:

[0038] A first processing module, configured to approximately map the Lindblad master equation to the newly constructed total Hamiltonian of the open quantum system with a stochastic differential equation as a bridge, where the newly constructed total Hamiltonian incorporates the system Hamiltonian of the open quantum system and the dissipation term of the interaction between the system and the environment;

[0039] A second processing module, configured to approximately transform the open quantum system into a new closed system with a stochastic differential equation as a bridge and derive the total Hamiltonian of the new closed system as the newly constructed total Hamiltonian, where the newly constructed total Hamiltonian incorporates the system Hamiltonian of the open quantum system and the dissipation term of the interaction between the system and the environment, and the dimension of the new closed system is higher than that of the open quantum system;

[0040] The third processing module is used to run a classical sampling method with Pauli operators as sampling samples, obtain the sampled Pauli operators, map the sampled Pauli operators into the combination of the quantum gates, construct a quantum circuit by using the combination of the quantum gates, and obtain the mean value of the target observable in the open quantum system according to the operation result of the quantum circuit.

[0041] In some embodiments, the second processing module is specifically configured to decompose the time evolution operator of the newly constructed total Hamiltonian into a linear superposition of the correction operator and the product of the Lie-Trotter formula by using the Lie-Trotter iteration decomposition and the Richardson extrapolation method, and represent the correction operator by the identity matrix and the local truncation error;

[0042] Decompose the local truncation error into a linear term combination of Pauli operators, and construct the combination of the quantum gates by using the coefficients in the linear term combination.

[0043] In some more specific embodiments, the second processing module is specifically configured to design a classical sampling algorithm and the corresponding quantum Monte Carlo circuit according to the newly constructed total Hamiltonian in combination with the leading term rotation method and the Richardson extrapolation method to obtain the mean value of any observable;

[0044] The principle of quantum simulation is based on:

[0045]

[0046]

[0047] and is the intermediate term for constructing ;

[0048] Among them, the 2m-order Lie-Trotter-Suzuki product formula is iteratively calculated according to the following formula:

[0049]

[0050] Among them, ,

[0051] ,

[0052] In the above formula, the leading term of its meaning is the lower-order term of , which is also the local truncation error, while is the higher-order term of the remaining is the identity matrix, is the identity matrix, is a parameter related to the order, is the order, is the correction operator, where, is the conjugate transpose of is the total Hamiltonian the unitary time evolution operator of

[0053] In some more specific embodiments, the second processing module is specifically configured to decompose the local truncation error into a linear combination of Pauli operators where is the first coefficient in the linear combination; is the first coefficient in the linear combination;

[0054] Calculate the sum of the absolute values of the first coefficient in the linear combination using the leading term rotation method ;

[0055] Determine the rotation angle through the sum of the absolute values of all coefficients , ;

[0056] Determine the second coefficient through the first coefficient and the rotation angle , ;

[0057] Construct a combination of quantum gates using the first coefficient , the second coefficient and the Pauli operator where is the quantum gate corresponding to the Pauli operator is the quantum gate corresponding to the Pauli operator , is the sign of the first coefficient and determines the rotation direction of the quantum gate.

[0058] In some embodiments, the third processing module is specifically configured to run a classical sampling method with Pauli operators as sampling samples on a classical computer, obtain the sampled Pauli operators, map the sampled Pauli operators into a combination of quantum gates, and obtain a quantum circuit;

[0059] Execute the quantum circuit on a quantum computer to obtain the expected value of the target observable;

[0060] Repeatedly execute the above operations, and determine the mean value of the target observable in the open quantum system according to the expected values obtained from multiple operation processes.

[0061] In some embodiments, the first processing module is specifically configured to perform a randomized approximation process on the Lindblad master equation to transform it into a stochastic differential equation;

[0062] Based on the stochastic differential equation, a newly constructed total Hamiltonian of the open quantum system is derived .

[0063] According to a third aspect, a computer storage medium is provided. A computer program is stored on the computer-readable storage medium. When the computer program is executed by one or more processors, the stochastic simulation method of the open quantum system as described in any one of the above technical solutions is implemented.

[0064] According to a fourth aspect, an electronic device is provided, including a memory and one or more processors. A computer program is stored on the memory. When the computer program is executed by the one or more processors, the stochastic simulation method of the open quantum system as described in any one of the above technical solutions is implemented.

[0065] In the above systems and methods provided in the embodiments of the present specification, a shallower quantum circuit is used to simulate the dynamic problems of large-scale actual quantum systems, alleviating the limitation of quantum chip resources. By using error mitigation techniques, the circuit depth is reduced, and part of the calculation process is allocated to a classical computer, while only the process involving observables is run on a quantum computer. Finally, the calculation results of the two parts are combined, saving valuable quantum hardware resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] To more clearly illustrate the technical solutions of the embodiments of the present application, the accompanying drawings required for the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0067] Figure 1 Shows a schematic diagram of directly mapping the Lindblad master equation provided in the embodiments of the present specification into a Schrödinger-like equation;

[0068] Figure 2 Shows a schematic diagram of the first block coding method provided in the embodiments of the present specification;

[0069] Figure 3 Shows a schematic diagram of the second block coding method provided in the embodiments of the present specification;

[0070] Figure 4 Shows a schematic diagram of the process of the stochastic simulation method of the quantum system provided in the embodiments of the present specification;

[0071] Figure 5 Schematic flowchart showing the random simulation method of an open quantum system provided by an embodiment of this specification;

[0072] Figure 6 Schematic diagram showing the random simulation device of an open quantum system provided by an embodiment of this specification. Detailed implementation manners

[0073] The following describes the solution provided by this specification in conjunction with the accompanying drawings.

[0074] In order to make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the following will describe the technical solutions in the embodiments of this application in conjunction with the accompanying drawings.

[0075] In the description of the embodiments of this application, words such as "exemplary", "for example", or "for illustration" are used to indicate examples, illustrations, or explanations. Any embodiment or design solution described as "exemplary", "for example", or "for illustration" in the embodiments of this application should not be construed as being more preferred or having more advantages than other embodiments or design solutions. Rather, the use of words such as "exemplary", "for example", or "for illustration" is intended to present relevant concepts in a specific manner.

[0076] In the description of the embodiments of this application, the term "and / or" is merely an association relationship describing associated objects, indicating that three relationships may exist. For example, A and / or B may represent: A exists alone, B exists alone, and A and B exist simultaneously. In addition, unless otherwise specified, the meaning of the term "plurality" refers to two or more.

[0077] In addition, the terms "first" and "second" are only used for descriptive purposes and cannot be construed as indicating or implying relative importance or implicitly indicating the technical features indicated. Thus, features defined with "first" and "second" may explicitly or implicitly include one or more of such features. The terms "include", "comprise", "have" and their variants all mean "including but not limited to", unless otherwise particularly emphasized in other ways.

[0078] Since in the quantum world, the quantum noise of the environment is inevitable, under the assumptions that a realistic open quantum system is weakly coupled to the environment and the environment is Markovian, this process can be described by the Lindblad master equation. In order to simulate an open quantum system, in one solution architecture, such as Figure 1As shown, the Lindblad master equation is directly mapped into a Schrödinger-like equation. By adding appropriate auxiliary qubits, this non-unitary process is constructed into a larger unitary process. After running for a period of time, a post-selection measurement is performed on the auxiliary system, so that the remaining main system satisfies the Lindblad master equation. However, in each step of the evolution, quantum resources will inevitably be lost with a certain probability, and the overall resources will be lost at an exponential rate proportional to the number of evolution steps \(n\). When the system is small, the evolution time is short, the sub-circuit is shallow, and the number of evolution steps is small, the resource loss is acceptable. When the system is large or the evolution time is long, the cumulative loss is unacceptable, making the simulation unable to continue.

[0079] One current solution is to use block encoding techniques to handle time-dependent Hamiltonian and jump operators. One encoding method is as Figure 2 shown, where the Kraus operators are decomposed into unitary operators that can be approximated by matrix exponentials, and the dissipation parameters in the Lindblad master equation are all encoded by these four matrices. This method is applied to the vectorized form of the Lindblad equation. However, it does not comprehensively characterize numerical or model errors. More importantly, this method requires the density matrix to be input as a -dimensional vector, where is the dimension of the Hilbert space. Another block encoding method is as Figure 3 shown, that is, the dissipation operator is directly encoded into the form shown in Figure 3 to form a large Hermitian Hamiltonian . A deeper quantum circuit can be built on a quantum computer to simulate the evolution of an open quantum system for a longer time. However, when performing the Schrödinger equation simulation, if based on the traditional direct simulation method using the Trotter formula, in each step, if we want to achieve precision, the circuit depth must be at least , and the circuit cost will be huge.

[0080] To solve the problem of circuit cost that occurred during the simulation of the Schrödinger equation above, in the prior art one, the Lindblad master equation needs to be mapped to a Schrödinger-like equation, then expanded into a high-dimensional Hermitian system by adding auxiliary qubits, and finally projected and numerically post-processed through post-selection to be re-converted into the Lindblad master equation. This type of solution requires calculating a relatively large matrix time-dependent integral in the middle, which is difficult to calculate when the system is large due to the high matrix dimension. And during the post-selection process, there is probability loss and resource consumption at each step, making it difficult to simulate for a long time. The prior art two is a solution based on Kraus decomposition. This type of solution usually takes the Kraus decomposition form of the master equation as a premise, which is a non-Hermitian matrix, then performs unitary expansion, and finally realizes the simulation by performing post-selection measurement on the auxiliary qubits. This type of solution requires performing eigenvalue decomposition on an intermediate matrix when decomposing the master equation into Kraus form, and the computational complexity increases exponentially with the system size, making it actually very difficult to perform and only applicable to very small systems. The prior art three is a solution based on variational quantum algorithms. This type of solution usually transforms the Lindblad master equation into a non-Hermitian linear differential equation, such as through vectorization or via the stochastic Schrödinger equation, etc., and then realizes the relevant state evolution through variational quantum algorithms. The main drawbacks of this type of solution include that a measurement process is required at each step, the process is cumbersome, and additional measurement errors may be introduced. When the system scale is large, due to the increase in the number of parameters to be solved, the "barren plateau problem" that often occurs in this algorithm may occur, resulting in the inability to continue the calculation.

[0081] To solve the above problems, as Figure 4 shown, based on the stochastic differential equation (SDE), the present invention maps the open quantum system simulation problem into a Hamiltonian simulation problem, uses the quantum circuit Monte Carlo (QCMC) technique, combines the leading term rotation method and the Richardson extrapolation method, combines the advantages of classical computers and quantum computers respectively, and designs a stochastic simulator for open quantum systems, which can achieve the simulation of open quantum systems with high precision. The solution of the inventive concept mainly includes: writing out the relevant Lindblad master equation according to the open quantum system to be simulated; performing a stochastic approximation process on the Lindblad master equation to convert it into a stochastic differential equation, and using the stochastic differential equation to deduce the Hamiltonian of the open quantum system ; according to the deduced , in combination with the leading-order-rotation method and the Richardson extrapolation method, design a classical sampling algorithm and the corresponding quantum Monte Carlo circuit. Run the classical sampling algorithm on a classical computer respectively, and run the quantum Monte Carlo circuit on a quantum computer according to the overall samples obtained on the classical computer to determine the mean value of the target observable in the open quantum system.

[0082] As Figure 5 shown, the stochastic simulation method of the open quantum system includes the following steps:

[0083] 110. Take the Lindblad master equation as a bridge with the stochastic differential equation, approximately transform the open quantum system into a new closed system, and derive the total Hamiltonian of the new closed system as the newly constructed total Hamiltonian. The newly constructed total Hamiltonian incorporates the system Hamiltonian of the open quantum system and the dissipative term of the interaction between the system and the environment. The dimension of the new closed system is higher than that of the open quantum system.

[0084] Specifically, the Lindblad master equation is:

[0085] The Lindblad master equation describes the evolution of the quantum state of the open quantum system over time. The density matrix contains all possible quantum state information of the quantum state of the open quantum system at time t. is the system Hamiltonian of the open quantum system, which describes the internal interactions of the open quantum system. describes the evolution of the open quantum system when there is no interaction with the external environment. describes the dissipation of the interaction between the open quantum system and the environment. is the j-th operator interacting with the environment. is the conjugate transpose of describes the decoherence effect caused by the interaction between the open quantum system and the environment.

[0086] Perform a randomized approximation process on the Lindblad master equation to become a stochastic differential equation, with an approximate accuracy of , the stochastic differential equation:

[0087]

[0088] In the conversion process, use for processing. In the formula, represents the wave function of the quantum state at time t. The open quantum system may be in a mixture of multiple pure states, and each pure state has a certain probability. The expectation value operator It is to calculate the statistical average effect of these pure states.

[0089] The stochastic differential equation describes the evolution of an open quantum system in a stochastic environment, representing the change of the quantum state of the open quantum system, describing the unitary evolution of the open quantum system without external perturbations, describing the dissipative term related to the interaction between the open quantum system and the environment, representing the deterministic evolution of the open quantum system caused by the Hamiltonian and the dissipative term. representing the stochastic evolution of the quantum state of the open quantum system due to the stochastic interaction with the environment, is the increment of the Brownian motion related to the j-th environment, used to simulate the influence of the environment on the open quantum system.

[0090] Write the stochastic differential equation in the Kraus form and calculate the Kraus operators ;

[0091]

[0092] It should be noted that here the stochastic differential equation is written in the Kraus form, and the equation in the Kraus form is approximate to the original stochastic differential equation, and no Kraus decomposition is performed.

[0093] By adding qubit A, find the evolution operator of the open quantum system , and obtain the state of the open quantum system:

[0094]

[0095] The above equation describes that after the open quantum system interacts with the environment, the density matrix is iteratively updated through unitary evolution and partial trace operation , where is the unitary evolution operator, representing the forward process of the evolution of the open quantum system at time , represents the reverse process of the evolution of the open quantum system at time ∆t, and the projection operator indicates that the auxiliary qubit is in the ground state, m represents the number of auxiliary qubits, and TrA represents the partial trace operation on subsystem A.

[0096] Derive the above equation to obtain the newly constructed total Hamiltonian of the open quantum system , the newly constructed total Hamiltonian of the open quantum system includes the system Hamiltonian H of the open quantum system, the Hamiltonian of the environment, and the interaction Hamiltonian between the open quantum system and the environment.

[0097] In this embodiment, by the above method, the problem that it is difficult to implement on a classical computer due to the extremely high dimension of the F matrix when mapping the Lindblad master equation into an F matrix during the process of solving the Lindblad master equation is avoided.

[0098] 120. Decompose the time evolution operator of the newly constructed total Hamiltonian into a linear combination of Pauli operators, and construct a combination of quantum gates using the coefficients of the linear combination. The combination of quantum gates is used to calculate the target observable in the open quantum system.

[0099] Specifically, using the Lie-Trotter iteration decomposition and Richardson extrapolation method, decompose the time evolution operator of the newly constructed total Hamiltonian into a linear superposition of the product of the correction operator and the Lie-Trotter formula, and represent the correction operator through the identity matrix and the local truncation error.

[0100] Specifically, according to the newly constructed total Hamiltonian , combine the leading term rotation method and the Richardson extrapolation method to design a classical sampling algorithm and the corresponding quantum Monte Carlo circuit to obtain the mean value of any observable;

[0101] The principle of quantum simulation is based on:

[0102]

[0103]

[0104] and is the intermediate term for constructing ;

[0105] Among them, the 2m-order Lie-Trotter-Suzuki product formula is iteratively calculated according to the following formula:

[0106]

[0107] Among them, ,

[0108] ,

[0109] In the above formula, the leading term of its meaning is the lower-order term of , which is also the local truncation error, while is the higher-order term of the remaining is the identity matrix, is the identity matrix, is a parameter related to the order, is the order, is the correction operator, where, is the conjugate transpose of, is the total Hamiltonian the unitary time evolution operator of at time t, and m is an integer.

[0110] Specifically, is the identity matrix, which is used to represent the part of the open quantum system that does not change during the evolution. The local truncation error represents the difference between the approximate unitary evolution operator and the time evolution operator of the total Hamiltonian, including the linear term and the high-order term .

[0111] Using Richardson extrapolation, the exponential order of the local truncation error is increased to a preset value.

[0112] is the low-order term of, which can be explicitly calculated, , is the remaining high-order term of.

[0113] Specifically, the local stage error is decomposed into a linear combination of Pauli operators, and a combination of quantum gates is constructed using the coefficients in the linear term combination, including:

[0114] Decompose the local truncation error into a linear combination of Pauli operators where is the first coefficient of the corresponding Pauli operator ; the first coefficient of the linear combination;

[0115] Decompose the linear term of the local truncation error into a linear combination of Pauli operators where is the first coefficient in the linear combination; ;

[0116] Calculate the sum of the absolute values of the first coefficients in the linear combination ;

[0117] Determine the rotation angle through the sum of the absolute values of all coefficients , ;

[0118] Determine the second coefficient through the first coefficient and the rotation angle , ;

[0119] Using the first coefficient , the second coefficient and the Pauli operator , construct a combination of quantum gates , where is the quantum gate corresponding to the Pauli operator , and is the sign of the first coefficient , which determines the rotation direction of the quantum gate.

[0120] 130. Run a classical sampling method with Pauli operators as sampling samples to obtain the sampled Pauli operators, map the sampled Pauli operators to a combination of quantum gates, construct a quantum circuit using the combination of quantum gates, and obtain the mean value of the target observable in the open quantum system according to the operation result of the quantum circuit.

[0121] Specifically, run a classical sampling method with Pauli operators as sampling samples on a classical computer to obtain the sampled Pauli operators, and map the sampled Pauli operators into a combination of quantum gates to obtain a quantum circuit;

[0122] Execute the quantum circuit on a quantum computer to obtain the expected value of the target observable;

[0123] Repeatedly execute the above operations, and determine the mean value of the target observable in the open quantum system according to the expected values obtained from multiple operation processes.

[0124] Specifically, the observable can be a physical quantity, such as magnetic field strength or energy intensity, etc. In the field of quantum chemistry, the stochastic simulation method of open quantum systems can be applied to simulate chemical molecular structures, chemical reactions, etc., to achieve more efficient and lower energy-consuming chemical design. When applied to chemical reactions, by using the Lindblad master equation as a bridge with a stochastic differential equation, approximately convert the chemical reaction as an open quantum system into a new closed system, and derive the total Hamiltonian of the new closed system as the newly constructed total Hamiltonian, where the newly constructed total Hamiltonian incorporates the system Hamiltonian of the open quantum system and the dissipative term of the interaction between the system and the environment. The dimension of the new closed system is higher than that of the open quantum system, and the dissipative term is the interaction of the environment with the chemical reaction. Finally, by simulating the chemical reaction, the physical and chemical data of the reaction products can be obtained.

[0125] Simulate molecular structures and electronic states, which are computationally intensive to simulate on traditional computers. VQE initializes a quantum computer to a default state and then transforms it into a desired (non-parameterized) reference state. It selects a parameterized quantum circuit (ansatz) to prepare the trial wave function and iteratively adjusts the circuit parameters to minimize the expected value of the Hamiltonian. This method effectively approximates the ground state energy of molecules in quantum chemistry simulations.

[0126] In this embodiment, through the above method, a shallower quantum circuit is used to simulate large-scale practical quantum system dynamics problems, alleviating the limitation of quantum chip resources. By using error mitigation techniques, the circuit depth is reduced, and part of the calculation process is allocated to a classical computer, while only the process involving observables is run on the quantum computer. Finally, the calculation results of the two parts are combined, saving valuable quantum hardware resources.

[0127] As Figure 5 shown, a stochastic simulation device for an open quantum system includes:

[0128] A first processing module for approximately transforming an open quantum system into a new closed system with the Lindblad master equation as a bridge to a stochastic differential equation, and deriving the total Hamiltonian of the new closed system as the newly constructed total Hamiltonian from it. The newly constructed total Hamiltonian incorporates the system Hamiltonian of the open quantum system and the dissipation term of the interaction between the system and the environment. The dimension of the new closed system is higher than that of the open quantum system;

[0129] A second processing module for decomposing the time evolution operator of the newly constructed total Hamiltonian into a linear combination of Pauli operators, and constructing a combination of quantum gates using the coefficients of the linear combination. The combination of quantum gates is used to calculate the target observable in the open quantum system;

[0130] A third processing module for running a classical sampling method with Pauli operators as sampling samples to obtain the sampled Pauli operators, mapping the sampled Pauli operators into the combination of quantum gates, constructing a quantum circuit using the combination of quantum gates, and obtaining the mean value of the target observable in the open quantum system according to the running result of the quantum circuit.

[0131] In some embodiments, the second processing module is specifically configured to decompose the time evolution operator of the newly constructed total Hamiltonian into a linear superposition of the product of a correction operator and the Lie-Trotter formula using the Lie-Trotter iterative decomposition and Richardson extrapolation method, and represent the correction operator through the identity matrix and the local truncation error;

[0132] Decompose the local truncation error into a linear combination of Pauli operators, and construct a combination of the quantum gates by using the coefficients in the linear combination.

[0133] In some more specific embodiments, the second processing module is specifically configured to, according to the newly constructed total Hamiltonian , combine the leading term rotation method and the Richardson extrapolation method to design a classical sampling algorithm and a corresponding quantum Monte Carlo circuit, and obtain the mean value of any observable.

[0134] The principle of quantum simulation is based on:

[0135]

[0136]

[0137] and is an intermediate term for constructing .

[0138] Among them, the 2m - order Lie - Trotter - Suzuki product formula is iteratively calculated according to the following formula:

[0139]

[0140] Among them, ,

[0141] ,

[0142] In the above formula, the leading term of its meaning is the lower - order term of, which is also the local truncation error, while , is the higher - order term of the remaining , is the identity matrix, is the identity matrix, is a parameter related to the order, is the order, is the correction operator. Among them, is the conjugate transpose of, is the total Hamiltonian the unitary time - evolution operator at time t, and m is an integer.

[0143] In some more specific embodiments, the second processing module is specifically configured to decompose the local truncation error into a linear combination of Pauli operators ​, where is the first coefficient in the linear combination;

[0144] Calculate the absolute value sum of the first coefficient in the linear combination using the leading term rotation method ;

[0145] Determine the rotation angle through the absolute value sum of all coefficients , , ;

[0146] Determine the second coefficient through the first coefficient and the rotation angle , , ;

[0147] Construct a combination of quantum gates using the first coefficient , the second coefficient and the Pauli operator , , is the quantum gate corresponding to the Pauli operator , is the sign of the first coefficient , determining the rotation direction of the quantum gate.

[0148] In some embodiments, the third processing module is specifically configured to run a classical sampling method with the Pauli operator as the sampling sample on a classical computer, obtain the sampled Pauli operator, map the sampled Pauli operator into a combination of quantum gates, and obtain a quantum circuit;

[0149] Execute the quantum circuit on a quantum computer to obtain the expectation value of the target observable;

[0150] Repeatedly execute the above operations, and determine the mean value of the target observable in the open quantum system according to the expectation values obtained from multiple operation processes.

[0151] In some embodiments, the first processing module is specifically configured to perform a randomized approximation process on the Lindblad master equation to become a stochastic differential equation;

[0152] Based on the stochastic differential equation, derive the newly constructed total Hamiltonian of the open quantum system.

[0153] The present invention also provides a computer storage medium, on which a computer program is stored. When the computer program is executed by one or more processors, it implements the stochastic simulation method for an open quantum system as described in any one of the above technical solutions.

[0154] The present invention also provides an electronic device, including a memory and one or more processors. A computer program is stored on the memory, and when the computer program is executed by the one or more processors, a random simulation method for an open quantum system as described in any one of the above technical solutions is implemented.

[0155] The specific embodiments described above further elaborate on the purpose, technical solutions, and beneficial effects of the present application. It should be understood that the above description is only the specific embodiments of the present application and is not used to limit the protection scope of the present application. Any modifications, equivalent replacements, improvements, etc. made on the basis of the technical solutions of the present application shall be included in the protection scope of the present application.

Claims

1. A stochastic simulation method for an open quantum system, characterized in that, The method includes: Taking the Lindblad master equation as a bridge with a stochastic differential equation, approximately transforming the open quantum system into a new closed system, and deriving the total Hamiltonian of the new closed system as the newly constructed total Hamiltonian therefrom, wherein the newly constructed total Hamiltonian incorporates the system Hamiltonian of the open quantum system and the dissipation term of the interaction between the system and the environment, and the dimension of the new closed system is higher than that of the open quantum system; Decomposing the time evolution operator of the newly constructed total Hamiltonian into a linear combination of Pauli operators, and constructing a combination of quantum gates using the coefficients of the linear combination, where the combination of quantum gates is used to calculate the target observable in the open quantum system; Running a classical sampling method with Pauli operators as sampling samples to obtain the sampled Pauli operators, mapping the sampled Pauli operators into the combination of quantum gates, constructing a quantum circuit using the combination of quantum gates, and obtaining the mean value of the target observable in the open quantum system according to the running result of the quantum circuit; The running of the classical sampling method with Pauli operators as sampling samples to obtain the sampled Pauli operators, mapping the sampled Pauli operators into the combination of quantum gates, constructing a quantum circuit using the combination of quantum gates, and obtaining the mean value of the target observable in the open quantum system according to the running result of the quantum circuit specifically includes: Running a classical sampling method with Pauli operators as sampling samples on a classical computer to obtain the sampled Pauli operators, and mapping the sampled Pauli operators into the combination of quantum gates to obtain a quantum circuit; Executing the quantum circuit on a quantum computer to obtain the expected value of the target observable; Repeatedly performing the above operations, and determining the mean value of the target observable in the open quantum system according to the expected values obtained from multiple operation processes.

2. The method according to claim 1, characterized in that, The decomposing the time evolution operator of the newly constructed total Hamiltonian into a linear combination of Pauli operators and constructing a combination of quantum gates using the coefficients of the linear combination specifically includes: Using the Lie-Trotter iteration decomposition and Richardson extrapolation method, decomposing the time evolution operator of the newly constructed total Hamiltonian into a linear superposition of a correction operator and the product of the Lie-Trotter formula, and expressing the correction operator through the identity matrix and the local truncation error; Decomposing the local truncation error into a linear combination of Pauli operators, and constructing the combination of quantum gates using the coefficients in the linear combination.

3. The method according to claim 2, wherein The using the Lie-Trotter iteration decomposition and Richardson extrapolation method to decompose the time evolution operator of the newly constructed total Hamiltonian into a linear superposition of a correction operator and the product of the Lie-Trotter formula, and expressing the correction operator through the identity matrix and the local truncation error specifically includes: According to the newly constructed total Hamiltonian Combining the principal term rotation method and the Richardson extrapolation method to design a classical sampling algorithm and the corresponding quantum Monte Carlo circuit to obtain the mean value of any observable quantity; The principle of quantum simulation is based on: K 2m (-Δt) + and K 2m (Δt) is an intermediate term for constructing S 2m ; Among them, the 2m-order Lie-Trotter-Suzuki product formula S 2m is iteratively calculated according to the following formula: S 2m = K 2m (-Δt) + K 2m (Δt) = e -iHΔt + O(Δt 2m+1 ) where K 2m (Δt) = K 2m-2 [(1 - 2rp r,m )Δt]S 2m-2 (p r,m Δt) r , K2(Δt) = S1(Δt / 2), p r,m = [2r - (2r) 1 / (2m+1) -1 ​ In the above formula, V 2m The leading term L of 2m (Δt) is 2m+1 O(Δt 2m ), which means the lower-order term of Δt and is also the local truncation error, while T 4m+2 (Δt) = Δt 2m is the higher-order term of the remaining Δt. I is the identity matrix, r is a parameter related to the order, m is the order, and V 2m (Δt) is the correction operator. Among them, K + 2m (-Δt) is the conjugate transpose of K 2m (Δt). is the unitary time evolution operator of the total Hamiltonian at time t. m is an integer, i is the imaginary unit, and K 2m (Δt) is a matrix. is the conjugate matrix of K 2m (Δt). K 2m (-Δt) + and are defined in the same way but with different representations. O is the infinitesimal symbol, and O(Δt 2m+1 ) means approximated to the order of magnitude of Δt 2m+1 . S1 is the first-order approximate representation of , and p r,m is the correlation coefficient.

4. The method according to claim 3, characterized in that, The decomposing the local truncation error into a linear combination of Pauli operators and constructing the combination of quantum gates using the coefficients in the linear combination specifically includes: Decompose the local truncation error L 2m into a linear combination of Pauli operators σ n such that L 2m = ∑ n a n · σ n , where a n is the first coefficient in the linear combination; Calculate the absolute value sum C of the first coefficients in the linear combination L = ∑ n |a n |; Sum of absolute values of all coefficients C L Determine the rotation angle φ, where φ = arctan C L ; Determine the second coefficient b through the first coefficient a n and the rotation angle φ n , Using the first coefficient a n , the second coefficient b n and the Pauli operator σ n , construct a combination of quantum gates where is the quantum gate corresponding to the Pauli operator σ n , sgn(a n ) is the sign of the first coefficient a n , which determines the rotation direction of the quantum gate.

5. The method according to claim 1, wherein Approximating the Lindblad master equation by using a stochastic differential equation as a bridge to transform an open quantum system into a new closed system, and deriving the total Hamiltonian of the new closed system as the newly constructed total Hamiltonian, specifically including: Performing a stochastic approximation on the Lindblad master equation to transform it into a stochastic differential equation; Approximating the transformation of the open system into a new closed system by adding auxiliary qubits; Derive the total Hamiltonian of the new closed system 6. A random simulation device for an open quantum system, characterized in that The device includes: A first processing module, configured to approximate the transformation of an open quantum system into a new closed system by using a stochastic differential equation as a bridge for the Lindblad master equation, and deriving the total Hamiltonian of the new closed system as the newly constructed total Hamiltonian, where the newly constructed total Hamiltonian incorporates the system Hamiltonian of the open quantum system and the dissipation term of the interaction between the system and the environment, and the dimension of the new closed system is higher than that of the open quantum system; A second processing module, configured to decompose the time evolution operator of the newly constructed total Hamiltonian into a linear combination of Pauli operators, and construct a combination of quantum gates by using the coefficients of the linear combination, where the combination of quantum gates is used to calculate the target observable in the open quantum system; A third processing module, configured to run a classical sampling method with Pauli operators as sampling samples to obtain the sampled Pauli operators, map the sampled Pauli operators into the combination of quantum gates, construct a quantum circuit by using the combination of quantum gates, and obtain the mean value of the target observable in the open quantum system according to the operation result of the quantum circuit; The third processing module is specifically configured to run a classical sampling method with Pauli operators as sampling samples on a classical computer to obtain the sampled Pauli operators, map the sampled Pauli operators into the combination of quantum gates to obtain a quantum circuit; Executing the quantum circuit on a quantum computer to obtain the expected value of the target observable; Repeatedly performing the above operations, and determining the mean value of the target observable in the open quantum system according to the expected values obtained from multiple operation processes.

7. The device according to claim 6, wherein: The second processing module is specifically configured to use Lie-Trotter iteration decomposition and Richardson extrapolation to decompose the time evolution operator of the newly constructed total Hamiltonian into a linear superposition of the product of a correction operator and the Lie-Trotter formula, and represent the correction operator by an identity matrix and a local truncation error; Decompose the local truncation error into a linear combination of Pauli operators, and construct the combination of quantum gates by using the coefficients in the linear combination.

8. A computer storage medium, characterized in that, A computer program is stored on the computer-readable storage medium, and when the computer program is executed by one or more processors, the stochastic simulation method of the open quantum system as described in any one of claims 1 to 5 is implemented.

9. An electronic device, characterized in that, Including a memory and one or more processors, a computer program is stored on the memory, and when the computer program is executed by the one or more processors, the stochastic simulation method of the open quantum system as described in any one of claims 1 to 5 is implemented.

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