Wave function optimization method and device, equipment and storage medium

Abstract

The embodiment of the invention relates to a wave function optimization method and device, equipment and a storage medium. The method proposed herein comprises: deriving a first heuristic wave function representation from a first wave function representation of the quantum system by reducing the energy of the quantum system in a parameter space corresponding to a wave function of the quantum system; determining, in a physical space corresponding to the quantum system, an estimate of the first reference quantum state by reducing an energy estimate corresponding to the first heuristic wave function representation; and updating the first wave function representation to a second wave function representation in a parameter space based on the first reference quantum state.

Description

TECHNICAL FIELD

[0001] Example embodiments of the present disclosure generally relate to the field of computer technology, and more particularly, to a wave function optimization method, apparatus, device and storage medium. BACKGROUND

[0002] Quantum chemistry is a computational science that studies the microscopic structure and reaction mechanism of matter based on the principles of quantum mechanics. The core task of quantum chemistry is to predict key parameters such as chemical bond properties, molecular configuration, reaction path and energy change of a quantum system based on the quantum behavior in the quantum system (a collection of particles at the atomic or molecular scale). In practical engineering applications of quantum chemistry, the wave function representation as the core data to describe the state of the quantum system is crucial for its computational efficiency and accuracy. Therefore, how to efficiently optimize the wave function representation to meet the actual application requirements is an urgent technical problem to be solved. SUMMARY

[0003] In a first aspect of the present disclosure, a wave function optimization method is provided. The method comprises: obtaining, in a parameter space corresponding to a wave function of a quantum system, a first trial wave function representation from a first wave function representation of the quantum system by lowering an energy of the quantum system; determining, in a physical space corresponding to the quantum system, an estimate of a first reference quantum state by lowering an energy estimate corresponding to the first trial wave function representation; and updating, in the parameter space, the first wave function representation to a second wave function representation based on the first reference quantum state.

[0004] In a second aspect of the present disclosure, an apparatus for wave function optimization is provided. The apparatus comprises: a trial wave function representation determination module configured to obtain, in a parameter space corresponding to a wave function of a quantum system, a first trial wave function representation from a first wave function representation of the quantum system by lowering an energy of the quantum system; a quantum state determination module configured to determine, in a physical space corresponding to the quantum system, an estimate of a first reference quantum state by lowering an energy estimate corresponding to the first trial wave function representation; and a wave function representation update module configured to update, in the parameter space, the first wave function representation to a second wave function representation based on the first reference quantum state.

[0005] In a third aspect of the present disclosure, an electronic device is provided. The device comprises at least one processor; and at least one memory coupled to the at least one processor and storing instructions for execution by the at least one processor. The instructions, when executed by the at least one processor, cause the device to perform the method of the first aspect.

[0006] In a fourth aspect of the present disclosure, a computer-readable storage medium is provided. The computer-readable storage medium has stored thereon a computer program, the computer program being executable by a processor to implement the method of the first aspect.

[0007] In a fifth aspect of the present disclosure, a computer program product is provided. The computer program product comprises computer executable instructions that, when executed by a processor, implement the method according to the first aspect of the present disclosure.

[0008] It should be understood that nothing in the Summary is to be construed as a limitation on the scope of the embodiments of the present disclosure. Other features of the present disclosure will be apparent from the following description, along with the accompanying figures and claims. BRIEF DESCRIPTION OF DRAWINGS

[0009] The above and other features, aspects, and advantages of embodiments of the present disclosure will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings. In the drawings, like reference numerals refer to like elements, wherein:

[0010] Figure 1 a schematic diagram showing an example environment in which embodiments of the present disclosure can be implemented;

[0011] Figure 2 a flowchart showing an example process of a wave function optimization method according to some embodiments of the present disclosure;

[0012] Figure 3 a schematic diagram showing an example of an optimized wave function representation according to some embodiments of the present disclosure;

[0013] Figure 4 a schematic block diagram showing an apparatus for wave function optimization according to some embodiments of the present disclosure; and

[0014] Figure 5 a block diagram of an electronic device in which one or more embodiments of the present disclosure can be implemented. DETAILED DESCRIPTION

[0015] Embodiments of the present disclosure will be described below in greater detail with reference to the accompanying drawings. While certain embodiments of the present disclosure are shown in the drawings, it is understood that the present disclosure can be embodied in various forms and should not be construed as being limited to the embodiments set forth herein; rather, these embodiments are provided so that the present disclosure will be more thoroughly and completely understood. It should be understood that the drawings and embodiments of the present disclosure are for illustrative purposes only and are not intended to limit the scope of the present disclosure.

[0016] It is to be noted that the headings provided herein are not limitations of various aspects of the disclosure. Various embodiments are described throughout this document and any type of embodiment can be included under any heading. Moreover, embodiments described in any heading can be combined with any other embodiment described in the same heading and / or a different heading in any manner.

[0017] In the description of embodiments of the disclosure, the term "includes" and its derivatives, such as "including" should be understood in an open way, i.e., "including but not limited to". The term "based on" should be understood as "at least partially based on". The term "one embodiment" or "the embodiment" should be understood as "at least one embodiment". The term "some embodiments" should be understood as "at least some embodiments". The following can also include other explicit and implicit definitions. The terms "first", "second", etc. can refer to different or same objects. The following can also include other explicit and implicit definitions.

[0018] In embodiments of the disclosure, data related to users, acquisition and / or use of data, etc. can be involved. These aspects comply with corresponding laws and regulations and relevant provisions. In embodiments of the disclosure, all data collection, acquisition, processing, processing, forwarding, use, etc. are carried out on the premise that the user is aware of and confirms. Accordingly, when implementing various embodiments of the disclosure, the type of data or information that can be involved, the scope of use, the scenario of use, etc. should be notified to the user and the authorization of the user should be obtained in a proper manner according to relevant laws and regulations. The specific notification and / or authorization manner can vary according to the actual situation and application scenario, and the scope of the disclosure is not limited in this respect.

[0019] In the specification and embodiments, if the scheme involves processing of personal information, it will be processed on the premise of legality (for example, obtaining the consent of the subject of personal information, or being necessary for the performance of a contract, etc.), and only within the scope of the provisions or agreement. The user refuses to process personal information other than the necessary information required for the basic function, which does not affect the user's use of the basic function.

[0020] As briefly described above, a core task of quantum chemistry is to predict key parameters such as chemical bond properties, molecular configurations, reaction paths, and energy changes based on quantum behavior at atomic and molecular scales. In one approach, simulation of a quantum system can be achieved by constructing and solving a many-electron Schrödinger equation.

[0021] The Schrödinger equation is the foundation for understanding the properties of molecules and crystals in physics and chemistry. Density Functional Theory (DFT) is the most widely used method for solving the Schrödinger equation, but it has limitations in approximating the exchange-correlation functional, which can lead to inaccurate results, especially in strongly correlated systems.

[0022] In contrast, wavefunction-based methods such as Coupled Cluster (CC) theory, Configuration Interaction (CI) theory, Density Matrix Renormalization Group (DMRG), and Quantum Monte Carlo (QMC) methods provide higher accuracy and have demonstrated effectiveness in a wide range of quantum chemistry problems. In practical applications, they still rely on various approximations such as finite basis sets, truncation of correlation effects, and specific forms of wavefunction trial states to avoid unaffordable computational costs. In this case, error cancellation is introduced. Error cancellation ensures the effectiveness of these methods in terms of accuracy and feasibility, but in more complex systems, especially strongly correlated systems or systems involving transition metals, error cancellation can become difficult to control and quantify. To alleviate this problem, multiple methods can be used simultaneously, and conclusions can only be drawn when different methods yield consistent results. However, this approach usually requires a large amount of computational resources, relies on expert knowledge to select and implement different methods, and requires careful analysis of results to reconcile possible inconsistencies.

[0023] Neural Network Quantum Monte Carlo (NNQMC) provides an alternative approach to achieve higher accuracy. Neural networks can effectively represent highly complex wave functions, thus accurately solving the Schrödinger equation for different systems. By the variational principle, the ground state of a quantum system can be rigorously determined by minimizing the energy in the Hilbert space of quantum states. Two key elements are required for an approximate ground state: an efficient optimization algorithm and a well-designed search space. NNQMC addresses both aspects simultaneously and becomes a high-precision approach. First, it adopts a powerful gradient optimization algorithm. Second, its search space is represented by a highly expressive neural network ansatz with tens or even hundreds of thousands of trainable parameters. NNQMC has achieved unprecedentedly low variational energies in systems such as nitrogen dimer and benzene dimer, demonstrating its ability to approximate the ground state. However, NNQMC still faces challenges: in strongly correlated or complex systems involving transition metals, variational optimization may fall into poor local minima, resulting in insufficient accuracy. At this time, simply expanding the size of the neural network to increase model complexity often exacerbates convergence difficulties and fails to improve accuracy.

[0024] Therefore, embodiments of the present disclosure provide a wave function optimization scheme. According to the scheme, first, in a parameter space corresponding to a wave function of a quantum system, a first trial wave function representation is obtained from a first wave function representation of the quantum system by reducing an energy of the quantum system. Then, in a physical space corresponding to the quantum system, an estimate of a first reference quantum state is determined by reducing an energy estimate corresponding to the first trial wave function representation. Subsequently, based on the first reference quantum state, the first wave function representation is updated to a second wave function representation in the parameter space.

[0025] It will be understood more clearly through the following description that, unlike the traditional NNQMC scheme, the scheme of the present disclosure extends the process of wave function optimization from the previous parameter space to the physical space of the quantum system, and realizes the progressive optimization of the wave function through a phased (parameter space and physical space) optimization strategy. In this way, the scheme of the present disclosure far exceeds the traditional NNQMC scheme in terms of wave function optimization accuracy. In addition, the scheme of the present disclosure can realize the law of neural scaling. That is, for a wave function representation constructed based on a neural network, the optimization accuracy follows a power law relationship with the expansion of the size of the neural network and the computing resources.

[0026] Compared with the local optimum of the traditional NNQMC scheme limited to a fixed parameter space, the scheme of the present disclosure breaks through the accuracy bottleneck of the traditional NNQMC scheme through the cooperative optimization of the physical space and the parameter space. Thus, the scheme of the present disclosure makes it possible to simulate quantum systems of strongly correlated systems, thus making it applicable to, for example, nitrogen dimer, ozone, benzene, and molecules containing transition metals, etc.

[0027] Various example implementations of the approach will be described in further detail below in conjunction with the following figures.

[0028] Figure 1 A schematic diagram illustrating an example environment 100 in which embodiments of the present disclosure can be implemented is shown. Referring to Figure 1 , the example environment 100 can include a terminal device 110 and an electronic device 120.

[0029] In the example environment 100, a user 130 can interact with the electronic device 120 via the terminal device 110 and / or its attached devices. As an example, the user 130 can provide the electronic device 120 with information related to a quantum system to be analyzed via the terminal device 110. After determining the quantum system to be analyzed, the electronic device 120 can establish a wave function representation of the quantum system and approximate the true ground state energy of the quantum system by iteratively optimizing the wave function representation to correspond to an energy estimate. After completing the optimization of the wave function representation, the electronic device 120 can determine various physical properties of the quantum system based on the optimized wave function representation and provide them to the user 130 for further analysis and processing.

[0030] In the example environment 100, the terminal device 110 can be any type of mobile terminal, fixed terminal, or portable terminal including a mobile handset, a tablet computer, a laptop computer, a notebook computer, a netbook computer, a tablet computer, a media computer, a multimedia tablet, a personal communication system (PCS) device, a personal navigation device, a personal digital assistant (PDA), an audio / video player, a digital camera / camcorder, a positioning device, a television receiver, a radio broadcast receiver, an electronic book device, a game device, or any combination thereof, including accessories and peripherals of these devices, or any combination thereof. In some embodiments, the terminal device 110 can also support any type of interface to a user (such as “wearable” circuitry, etc.).

[0031] The electronic device 120 can be a standalone physical server, a server cluster or a distributed system composed of multiple physical servers, or a cloud server providing cloud services, cloud databases, cloud computing, cloud functions, cloud storage, network services, cloud communications, middleware services, domain name services, security services, content distribution networks, and basic cloud computing services such as big data and artificial intelligence platforms. The electronic device 120 may, for example, include a computing system / server, such as a mainframe, an edge computing node, a computing device in a cloud environment, etc.

[0032] A communication connection may be established between the electronic device 120 and the terminal device 110. The communication connection may be established via a wired or wireless method. The communication connection may include, but is not limited to, a Bluetooth connection, a mobile network connection, a Universal Serial Bus connection, a Wi-Fi connection, etc., and the embodiments of the present disclosure are not limited in this respect. In the embodiments of the present disclosure, the electronic device 120 and the terminal device 110 may implement signaling interaction via the communication connection between them.

[0033] It should be understood that the structure and function of the various elements in the environment 100 are described for illustrative purposes only and do not imply any limitation on the scope of the present disclosure.

[0034] Figure 2 FIG. 2 is a flow chart illustrating an example process 200 of a wave function optimization method according to some embodiments of the present disclosure. The process 200 may be implemented at the electronic device 120 .

[0035] Reference Figure 2 In block 210 , the electronic device 120 obtains a first tentative wave function representation from the first wave function representation of the quantum system by reducing the energy of the quantum system in a parameter space corresponding to the wave function of the quantum system.

[0036] A quantum system can refer to a microscopic system described based on quantum mechanics. The wave function is a description of the state of the quantum system, which can indicate the probability distribution of particles in space. As an example, a quantum system can be described by the Schrödinger equation, which can be specifically expressed by formula (1):

[0037]

[0038] in is the energy operator of the quantum system (such as the Hamiltonian operator), ψ is the wave function representation, ψ(x1,x2,…,x N ) is the sample set (x1, x2,…, x N ) is represented by the wave function, x i For the sample set (x1, x2, ..., x N ) in the i-th sample (such as electron, etc.), i∈{1,2,…,N}, N is a positive integer, sample x i ={r i ,σ i},in Represents sample x i The position in the quantum system, σ i ∈{+1,-1} represents sample x i The spin state of the quantum system is E, and E is the ground state energy of the quantum system. Such a wave function represents ψ(x1,x2,…,x N) can also be referred to as a many-electron wave function representation. Based on the spin-statistics theorem, the many-electron wave function satisfies an antisymmetry when exchanging any two electrons.

[0039] As an example, the energy operator can be represented by equation (2):

[0040]

[0041] where is a Laplacian operator, represents a sample x j is a position of the j-th nucleus in the quantum system, j e {1, 2, …, N}, Z I is a charge number of the I-th nucleus (I e {1, 2, …, M}, M is a positive integer), R I is a position of the I-th nucleus in the quantum system, Z J is a charge number of the J-th nucleus (J e {1, 2, …, M}, M is a positive integer), R J is a position of the J-th nucleus in the quantum system.

[0042] In some embodiments, the wave function representation can be determined based on a neural network or other means. The wave function representation determined based on a neural network has an expression form of a neural network, and thus such a wave function representation can also be referred to as a neural network wave function representation. For convenience of discussion, unless otherwise specified, the wave function representation referred to below can refer to a neural network wave function representation.

[0043] In some embodiments, the electronic device 120 optimizes the wave function representation of the quantum system through multiple iterations to make the energy of the quantum system approximate the ground state energy. The process shown in block 210 and the processes shown in blocks 220 and 230 to be mentioned below can be any one of these iterations. For convenience of discussion, the various embodiments of the present disclosure are described below with the process shown in blocks 210 to 230 as an example of the first iteration of these iterations. In this scenario, the first wave function representation can be understood as an initial wave function representation in the wave function optimization process. As an example, the first wave function representation can be constructed based on a neural network, and such a first wave function representation can also be referred to as a neural network wave function ansatz.

[0044] The parameter space can refer to a set of adjustable parameters in the wave function representation. As an example, in a neural network wave function representation, the adjustable parameters can include network weights, bias terms, and hidden layer structures, etc. As an example, a wave function representation ψ θ (x) can be represented by equation (3):

[0045]

[0046] where J θ (x) is an inter-electronic correlation function (also referred to as a Jastrow factor), is a sum over Slater Determinants, k e {1, 2, …, K}, K is a positive integer, each determinant corresponds to one determined by a neural network, and 0 is an adjustable parameter of the wave function representation.

[0047] In some embodiments, the electronic device 120 can reduce the energy of the quantum system (here, the energy can refer to the total energy of the quantum system) by adjusting the adjustable parameter of the first wave function representation in the parameter space. In some embodiments, the electronic device 120 can adjust the first wave function representation based on a Monte Carlo algorithm until the energy of the quantum system is reduced to meet the energy requirement. Then, the electronic device 120 determines the first trial wave function representation based on the adjusted first wave function representation.

[0048] As an example, the energy requirement can be a pre-set energy threshold or criterion. For example, the energy requirement can indicate that the electronic device 120 minimizes the energy of the quantum system. This means that in this process, the electronic device 120 will adjust the adjustable parameter of the first wave function representation to find the ground state energy of the quantum system. As an example, the electronic device 120 can adjust the adjustable parameter of the first wave function representation based on a variational Monte Carlo (VMC) algorithm. Once the energy of the quantum system meets the energy requirement, the electronic device 120 can determine that the adjusted wave function representation meets the expectation, thereby obtaining the first trial wave function representation.

[0049] More details of VMC in the embodiments of the present disclosure are described below in conjunction with equations (4) to (7).

[0050] As an example, VMC can adjust the adjustable parameter 0 of the wave function representation to approximate the ground state energy of the quantum system by solving an optimization problem, which can be represented by equation (4):

[0051]

[0052] where the adjustable parameter 0 can also be referred to as ansatz parameters.

[0053] As an example, the wave function representation satisfies the anti-symmetry condition, which can be represented by equation (5):

[0054]

[0055] As an example, the process of VMC calculating the total energy E of a quantum system can be represented by formula (6): tot

[0056]

[0057] where E L is the local energy of a sample in the quantum system, which can be represented by formula (7):

[0058]

[0059] By VMC, the electronic device 120 can quickly find the wave function representation corresponding to the ground state energy of the quantum system in the parameter space, thereby obtaining the first trial wave function representation.

[0060] To more accurately obtain the first trial wave function representation, in some embodiments of the present disclosure, a Kronecker factorization approximation curvature can be further introduced in the VMC. Specifically, the electronic device 120 determines the local energies of a plurality of third samples in the quantum system based on the first wave function representation and the energy operator of the quantum system. Then, the electronic device 120 determines a second energy gradient for the first wave function representation based on the local energies of the plurality of third samples and the total energy of the quantum system corresponding to the first wave function representation. Subsequently, the electronic device 120 adjusts the first wave function representation based on the second energy gradient adjusted by the Kronecker factorization approximation curvature to determine the first trial wave function representation.

[0061] The energy operator of the quantum system can be a Hamiltonian, which can indicate the kinetic energy term and potential energy term of the quantum system, etc. The plurality of third samples can be samples corresponding to the first wave function representation in the quantum system determined by the electronic device 120 based on the Monte Carlo sampling algorithm. The local energies of the plurality of third samples can be energy values calculated by the electronic device 120 for each third sample based on the first wave function representation and the energy operator. The total energy of the quantum system corresponding to the first wave function representation can be obtained by averaging the local energies of the plurality of third samples by the electronic device 120. The second energy gradient can indicate the change direction of the energy of the quantum system, which is used to guide the electronic device 120 to increase or decrease the adjustable parameters of the first wave function representation.

[0062] The Kronecker factorization approximation curvature is used to further optimize the above process. Specifically, the Kronecker factorization approximation curvature accelerates the convergence speed of the above process and improves the quality of the solution by approximating the Hessian matrix, so that it can effectively handle complex first wave function representations, especially in high-dimensional space to find the optimal solution. ​

[0063] The following describes more details of how the electronic device 120 determines the representation of the first trial wave function in conjunction with formulas (8) to (11).

[0064] As an example, the process of the electronic device 120 determining the local energy of the third sample can be expressed by formula (8):

[0065]

[0066] where ψ θ 9x (i) ) is for sample x (i) The wave function with adjustable parameter θ is represented by, for example, n The first wave function of E L (x (i) ) is based on the wave function representation ψ θ (x (i) ) calculated quantum system sample x (i) The local energy.

[0067] As an example, the process of the electronic device 120 determining the total energy of the quantum system can be expressed by formula (9):

[0068]

[0069] Where B is the batch size, for example, the total energy E of the quantum system is calculated each time tot The total number of samples in the sample set.

[0070] As an example, the second energy gradient g can be expressed by formula (10):

[0071]

[0072] where g is the second energy gradient, is relative to the adjustable parameter θ n The gradient operation, which can be based on the loss function or the objective function with respect to the adjustable parameter θ n Determined by the derivative of For sample x (i) with adjustable parameter θ n ’s wave function representation, such as the first wave function representation.

[0073] As an example, the electronic device 120 adjusts the adjustable parameter θ represented by the first wave function n , the process of determining the first trial wave function representation can be expressed by formula (11):

[0074]

[0075] where θ' is the adjusted adjustable parameter, e.g., the adjustable parameter corresponding to the first trial wave function representation, and τ is a learnable parameter of the gradient descent algorithm. n+1 where θ' is the adjusted adjustable parameter, e.g., the adjustable parameter corresponding to the first trial wave function representation, and τ is a learnable parameter of the gradient descent algorithm. is a Kronecker factorization approximation of the curvature.

[0076] After determining the first trial wave function representation, the method 200 proceeds to block 220. At block 220, the electronic device 120 determines an estimate of a first reference quantum state by reducing an energy estimate corresponding to the first trial wave function representation in a physical space corresponding to the quantum system.

[0077] The physical space can be a space that describes the state of the quantum system in the real world (e.g., the positions, shapes, sizes, etc. of the individual samples). Such a physical space can be represented, for example, by a Hilbert space. Unlike the parameter space, which focuses on the mathematical parameter level of description, the physical space is directly related to the actual existence and movement of the samples in the quantum system.

[0078] The energy estimate corresponding to the first trial wave function representation can refer to the total energy of the quantum system determined by the electronic device 120 based on the first trial wave function representation. As an example, the electronic device 120 can adjust (one or more times) the first trial wave function representation in the physical space and find a better direction (e.g., capable of reducing the energy estimate corresponding to the first trial wave function representation to the lowest energy of the quantum system) from the trial results to reduce the energy estimate corresponding to the first trial wave function representation. This process can be understood as further approximating the true ground state energy of the quantum system outside the parameter space through a trial strategy.

[0079] The quantum state can be a description of the state of the quantum system at a certain time, for example, the quantum state can indicate the probability information of all observable properties of the quantum system. As an example, the electronic device 120 can determine the quantum state corresponding to the lowest energy estimate determined in the above trial process as the first reference quantum state.

[0080] At block 230, the electronic device 120 updates the first wave function representation to a second wave function representation in the parameter space based on the first reference quantum state. As an example, the electronic device 120 can project the first reference quantum state to the parameter space through a projection mechanism, thereby determining a target parameter in the parameter space that best matches the first reference quantum state. Further, the electronic device 120 updates the first wave function representation to the second wave function representation by adjusting the adjustable parameters of the first wave function representation towards the target parameter.

[0081] Embodiments of the present disclosure expand the process of wave function optimization from the parameter space to the physical space, so that the lowest energy of the quantum system can be explored in a wider range during the process of wave function optimization, thereby increasing the possibility of finding a global optimal solution. By searching in the physical space, the electronic device 120 can get rid of the restrictions of the parameter space, and thus effectively avoid the problem of wave function optimization falling into a local minimum value.

[0082] In embodiments of the present disclosure, the processes shown in blocks 220 and 230 can also be understood as introducing a Lookahead mechanism on the variational algorithm involved in block 210. Therefore, the processes shown in blocks 210 to 230 can also be referred to as a LookAhead Variational Algorithm (LAVA). In embodiments of the present disclosure, LAVA aims to optimize the wave function representation of the quantum system by combining Imaginary Time Evolution (ITE) and variational optimization. LAVA operates in an inner loop following the ITE trajectory, and iteratively updates the wave function representation through the projection mechanism of the outer loop, ensuring that the optimization result is limited within the parameter space.

[0083] Figure 3 A schematic diagram of an example 300 of LAVA optimizing a wave function representation is shown according to some embodiments of the present disclosure. More details of LAVA are described below in combination with Figure 3 a traditional Neural Network-based Variational Monte Carlo (NNVMC) as a comparative example.

[0084] In the traditional NNVMC, the wave function representation ψ can be decomposed based on the eigenfunctions of the energy operator : With time evolution, the wave function representation ψ tends to the ground state wave function representation ψ0, following the formula: In the formula, E0 is the ground state energy of the quantum system, and the eigenvalues {Ei} are arranged in ascending order, and c i is the reference coefficient. Starting from the wave function representation , NNQMC can be regarded as a process of iteratively updating the adjustable parameters θ i along the discretized ITE trajectory in the parameter space. In each iteration, NNQMC calculates a target wave function representation where E T is a scalar for normalization, and the target wave function representation Projected into the parameter space. As the number of iterations N increases, if the neural network wave function is assumed to perfectly express the discretized ITE trajectory, when NΔτ→+∞, then ψ θ →ψ 0. However, in practice this is not always possible due to the limited expressive power of the hypothesis.

[0085] Reference Figure 3 In the embodiment of the present disclosure, LAVA can first use VMC combined with the gradient descent algorithm (GD) to adjust the wave function representation in the parameter space 310. The adjustment of the wave function representation can approximate the target wave function representation. Then we get the adjusted adjustable parameter θ′ n+1 This process can also be called the first step 311 of wave function optimization. For details, please refer to the above formulas (10) and (11), so it will not be repeated here.

[0086] Next, LAVA does not directly represent the target wave function Instead of projecting it into the parameter space 310, the target wave function representation is adjusted tentatively to preview the target wave function representation after one or more steps (e.g. ). Then, LAVA moves along a tentatively found preferred direction (such as pointing to This process can also be called the second step 312 of wave function optimization. After completing the second step 312, LAVA projects the end point 313 of the second step 312 (for example, the first reference quantum state) into the parameter space 310, thereby searching for the target parameter 315 that best matches the first reference quantum state in the parameter space 310. This process can also be called the projection step 314. The above is roughly equivalent to taking a small step closer to LAVA then adjusts the adjustable parameters 316 of the first wave function representation toward the target parameters 315 in parameter space 310, thereby obtaining a second wave function representation. This process can also be referred to as the correction optimization step 317 of the first wave function representation. This strategy can be viewed as a "lead compensation" for traditional single-step ITE, allowing it to efficiently approximate the true ground state energy of a quantum system.

[0087] As an example, the wave function corresponding to the end point of the second step, ψ′(x), can be expressed by formula (12):

[0088]

[0089] in is a variable with adjustable parameters θ for sample x i-1 The wave function representation of (for example, the first wave function representation), is the wave function representation corresponding to the end point of the first step (for example, the first trial wave function representation), and C is the weight for the first trial wave function representation.

[0090] In some embodiments, LAVA can update the first wave function representation to the second wave function representation based on the score matching (SM) algorithm. As an example, LAVA can determine the direction of the decrease in score-matching loss as the direction of change of the first wave function representation. This process can be expressed by formula (13):

[0091]

[0092] in is the first gradient operator with respect to the reference variable r, is the probability density function, for example is a variable with adjustable parameters θ for sample x i ’s wave function representation (e.g., the second trial wave function representation).

[0093] As an example, the gradient of the change from the first wave function representation to the second wave function representation can be expressed by formula (14):

[0094]

[0095] Where sgn is the sign function, E′ L (x) represents the local energy of the sample x calculated based on the wave function representation ψ′(x), is about the adjustable parameter θ i The second gradient operator.

[0096] In some embodiments, electronic device 120 determines a combined wave function representation based on the first wave function representation and the first trial wave function representation. Then, electronic device 120 determines the local energies of multiple first samples of the quantum system based on the combined wave function representation and an energy operator for the quantum system. Subsequently, electronic device 120 determines an estimate of a first reference quantum state based on the local energies of the multiple first samples. In this way, electronic device 120 can comprehensively determine the first reference quantum state by combining the first wave function representation and the first trial wave function representation, thereby better approximating the true ground state wave function representation in physical space.

[0097] The combined wave function representation can be generated by the electronic device 120 linearly or nonlinearly combining the first wave function representation and the first trial wave function representation. The energy operator of the quantum system can refer to a Hamiltonian operator of the quantum system. For example, the energy operator can include a kinetic energy term, an electron-electron Coulomb repulsion term, and / or an electron-nucleus attraction term, etc. The plurality of first samples can be samples corresponding to the combined wave function representation determined by the electronic device 120 in the quantum system based on a Monte Carlo sampling method. For example, the plurality of first samples can be samples corresponding to the first trial wave function representation determined by the electronic device 120 in the quantum system based on a Monte Carlo sampling method.

[0098] As an example, the determination process of the plurality of first samples can be represented by formula (15):

[0099]

[0100] wherein is a wave function representation with an adjustable parameter θ' of the sample x, n+1 is a wave function representation with an adjustable parameter θ' of the sample x, n+1 The two wave function representations can be, for example, the first trial wave function representation.

[0101] For each first sample, the electronic device 120 can calculate a value of the energy operator acting on the combined wave function representation at the first sample. Then, the electronic device 120 determines a local energy of the first sample based on a ratio of the value to the combined wave function representation itself.

[0102] In some embodiments, the electronic device 120 determines a weight for the first trial wave function representation based on an inner product of the first wave function representation and an inner product of the first trial wave function representation. Then, the electronic device 120 determines the combined wave function representation based on the first wave function representation and the first trial wave function representation adjusted by the weight.

[0103] As an example, the electronic device 120 can determine the weight for the first trial wave function representation based on a ratio of the inner product of the first wave function representation to the inner product of the first trial wave function representation, etc. Such a weight can also be referred to as a normalization coefficient of the first trial wave function representation. In this way, the electronic device 120 can ensure that the proportion relationship of the first wave function representation and the first trial wave function representation in the combined wave function representation is correct.

[0104] As an example, the weight for the first trial wave function representation can be represented by formula (16):

[0105]

[0106] wherein ​is a variable with adjustable parameter θ n ’s wave function representation, such as the first wave function representation.

[0107] As an example, determining the local energy of the first sample can be expressed by formula (17):

[0108]

[0109] in For sample x (i) with adjustable parameter θ n The wave function representation of is a variable with adjustable parameters θ for sample x n The wave function representation of For sample x (i) with adjustable parameter θ′ n+1 The wave function representation, for example, the first trial wave function representation, E′ L (x (i) ) is based on the wave function representation The calculated local energy of sample i in the quantum system. It should be noted that and Both can indicate the first wave function representation, but the samples they target can be different. For example, sample x (i) It can be determined by updating the sample x based on formula (15).

[0110] Once the local energies of the plurality of first samples are determined, the electronic device 120 may determine an energy estimate corresponding to the first trial wave function representation based on an average value of the local energies, etc. Furthermore, the electronic device 120 may employ the aforementioned trial strategy to tentatively adjust the first trial wave function representation in physical space to reduce the energy estimate corresponding to the first trial wave function representation until the first reference quantum state is determined.

[0111] In some embodiments, the first wave function representation corresponds to a first parameter in parameter space 310 (e.g., adjustable parameter 316), and the electronic device 120 determines a parameter gradient at the first parameter in parameter space 310 based on the first reference quantum state (e.g., the gradient corresponding to the modified optimization step 317). Subsequently, the electronic device 120 adjusts the first parameter to a second parameter corresponding to the second wave function representation in parameter space 310 (e.g., target parameter 315) based on the parameter gradient.

[0112] The parameter gradient can be represented by a vector, a matrix, or other forms. The parameter gradient can instruct the electronic device 120 to increase or decrease the first parameter in the parameter space 310 to adjust the first parameter to a second parameter corresponding to the second wave function. The parameter gradient determined by the electronic device 120 based on the first reference quantum state can guide the electronic device 120 to correct the gradient direction in the parameter space 310 (relative to the gradient direction calculated by the traditional VCM, for example Figure 3 The first step 311 in the above method is performed to make the optimized wave function representation (eg, the second wave function representation) closer to the true ground state energy of the quantum system.

[0113] In some embodiments, electronic device 120 determines a first energy gradient for the first wave function representation based at least on a local energy of the quantum system corresponding to the first reference quantum state and a relative relationship between the local energy and a reference variable. Then, electronic device 120 converts the first energy gradient into a parameter gradient.

[0114] As an example, the local energy corresponding to the first reference quantum state may be the local energy of multiple samples in the quantum system when the quantum system reaches the first reference quantum state, such as the local energy E′ L (x (i) ). The reference variable can be the local energy E′ L (x (i) ) related to one or more variables, which can be determined according to actual needs. For example, the reference variable can indicate the sample x in the quantum system (i) The position r in the quantum system (i) Etc. The first energy gradient can be represented by a vector, a matrix or other forms. As an example, the local energy E′ L (x (i) ) relative to the reference variable can indicate the local energy E′ L (x (i) ) at position r (i) The gradient at , which can reflect the spatial rate of change of energy density. The first energy gradient can instruct the electronic device 120 to adjust the adjustable parameters represented by the first wave function so that the energy estimate represented by the first wave function approaches the true ground state energy of the quantum system. In some embodiments, after determining the first energy gradient, the electronic device 120 can use an optimizer to convert the first energy gradient into a parameter gradient. For example, the electronic device 120 can input the first energy gradient into an optimizer based on Adaptive Moment Estimation (Adam) and KFAC to convert the first energy gradient into a parameter gradient.

[0115] In this way, the electronic device 120 can efficiently convert the energy gradient taking into account the reference variable into a parameter adjustment strategy, thereby improving the accuracy and convergence speed of wave function parameter optimization.

[0116] In some embodiments, electronic device 120 determines a first energy gradient for the first wave function representation based on a local energy of the quantum system corresponding to the first reference quantum state, a relative relationship between the local energy and a reference variable, and a sensitivity of the first wave function representation to a parameter adjustment amplitude. The parameter adjustment amplitude refers to the degree of adjustment to a parameter of the wave function representation.

[0117] The sensitivity of the first wave function representation to the parameter adjustment amplitude can indicate the impact of changes in the adjustable parameter θ on the first wave function representation. For example, if a small change in the adjustable parameter θ causes a significant change in the first wave function representation, then the first wave function representation is highly sensitive to the parameter adjustment amplitude; conversely, the sensitivity is low. As an example, electronic device 120 can assess the sensitivity using second-order or higher-order derivatives.

[0118] In this way, the electronic device 120 is able to quantify the influence of these reference variables on the wave function optimization, so that the first energy gradient can be estimated more accurately.

[0119] The first energy gradient is described in more detail below in conjunction with formulas (18) to (23).

[0120] As an example, the first energy reduction gradient It can be expressed by formula (18):

[0121]

[0122] in is the local energy E′ L (x (i) ) for the reference variable r (i) The gradient, is a double gradient, which can indicate the tuning parameter θ n The amplitude of the change in the wave function ψ θ (x (i) ) at position r (i) The degree of influence of the gradient at .

[0123] As an example, the optimizer can adjust the first parameter to the second parameter directly based on the first energy gradient g. This process can be expressed by formulas (19) to (23):

[0124]

[0125] where β1, β2 and η are configurable parameters of Adam, ∈ is a smoothing term, m n is a first-moment estimate, v n is a second-moment estimate, and is a corrected moment estimate, is a parameter gradient. As an example, β1= 0.9 or other values, β2= 0.99 or other values, η can indicate a learning rate controlled by a scheduler.

[0126] In some embodiments, at least one of the first wave function representation, the second wave function representation, and the first trial wave function is determined based on a target neural network. An amount of parameters of the target neural network can be determined based on a target accuracy of the wave function optimization. For example, the electronic device 120 determines local energy variances of a plurality of second samples of the quantum system based on wave function representations output by a plurality of reference neural networks having different amounts of parameters. Then, the electronic device 120 fits a dependent variable of the target neural network based on the local energy variances. Subsequently, the electronic device 120 determines the amount of parameters of the target neural network based on the dependent variable and the target accuracy of the wave function optimization.

[0127] In some embodiments, the first wave function representation, the second wave function representation, and the first trial wave function are all determined based on a target neural network, which can also be referred to as a neural network wave function representation. The details of the neural network wave function representation are described above and will not be repeated here.

[0128] In embodiments of the present disclosure, LAVA can also implement a neural network scaling law between wave function optimization accuracy and neural network size. The inventors have proven through experiments that LAVA makes the energy error (e.g., the absolute difference from the true ground state energy) of the optimized wave function and the parameter size (e.g., the number of layers, the number of hidden units, or the complexity of the activation function) of the neural network have a clear mathematical relationship, such as a power law relationship. Based on this, embodiments of the present disclosure can determine an appropriate neural network size in advance according to the target accuracy of the wave function optimization in the actual application process of the wave function optimization, thereby achieving efficient use of resources.

[0129] In some embodiments, the plurality of reference neural networks having different amounts of parameters can be predetermined, for example, these reference neural networks can differ in the number of neurons, the number of network layers, etc. As an example, the electronic device 120 can optimize the wave function representations output by these neural networks through the processes shown in blocks 210 to 230.

[0130] For a plurality of second samples of the quantum system, the electronic device 120 calculates the local energy of each sample using the optimized wave function representation output by the reference neural network. Then, the electronic device 120 calculates the variance of these local energies. The local energy variance reflects the stability and accuracy of the wave function representation under different samples, and the smaller the variance, the more stable the wave function representation and the more accurately it describes the state of the quantum system. After determining the parameter quantity of the reference neural network and the local energy variance, the electronic device 120 can use a suitable fitting method (such as linear regression, polynomial fitting, neural network fitting, etc.) to fit the dependent variable of the target neural network, thereby obtaining the specific value of the dependent variable of the target neural network.

[0131] The target accuracy represents the degree of accuracy that the wave function representation is expected to optimize to. According to the relationship between the dependent variable obtained by fitting and the parameter quantity, and the target accuracy set, the electronic device 120 can automatically select the parameter quantity of the target neural network. For example, if the target accuracy requirement is high, the electronic device 120 can select a larger parameter quantity to obtain a more accurate wave function representation; on the contrary, if the target accuracy requirement is low, the electronic device 120 can appropriately reduce the parameter quantity to reduce the calculation cost.

[0132] As an example, the process of the electronic device 120 fitting the dependent variable of the target neural network can be represented by formula (24):

[0133]

[0134] where α v and β v are the dependent variable of the target neural network, V is the variance of the local energy, N p is the parameter quantity of the target neural network.

[0135] According to the zero variance principle of quantum Monte Carlo, when V→0, the estimated value of the local energy will converge to the exact ground state energy. Since the variance estimation does not need to rely on accurate reference data (unlike error measurement), this relationship holds for any system that can perform LAVA calculation. The inventors verified this law through electronic structure calculations on a variety of molecular systems, with an average R 2 value of more than 0.97. Although the generalized linear model may be more suitable for non-constant variance data, this process still uses OLS regression, as no significant heteroscedasticity is found in the residual diagnosis.

[0136] It should be noted that the various formulas and parameters involved in the present disclosure are only used as examples, and do not constitute a limitation on the embodiments of the present disclosure. According to actual needs, the embodiments of the present disclosure can also use other formulas and parameters, etc.

[0137] As described previously, embodiments of the present disclosure can optimize the wave function representation through multiple iterations. As described previously, the process shown in blocks 210-230 can be considered as any one of the multiple iterations (e.g., the first iteration). In some embodiments, the multiple iterations further include a second iteration after the first iteration. Specifically, in the second iteration, the electronic device 120 obtains a second trial wave function representation from the second wave function representation by lowering the energy of the quantum system in the parameter space 310. Then, the electronic device 120 determines an estimate of the second reference quantum state by lowering the energy estimate corresponding to the second trial wave function representation in the physical space. Next, the electronic device 120 updates the second wave function representation to a third wave function representation in the parameter space 310 based on the second reference quantum state. In turn, in response to the third wave function representation satisfying the convergence condition, the electronic device 120 determines the physical property of the quantum system based on the third wave function representation.

[0138] The convergence condition can be a criterion used to determine whether the wave function representation has met the expectation and thus the iteration can be stopped. The convergence condition can include that the energy change is less than a certain threshold (i.e., the difference between the energy estimates obtained by two adjacent iterations is less than a preset value), the change of the wave function parameters is less than a certain threshold, etc. For example, the electronic device 120 can compare the third wave function representation with the wave function representation obtained in the previous iteration (e.g., the second wave function representation), so as to check whether the convergence condition is met. If the convergence condition is met, it means that the wave function representation has met the expectation, and the iteration can be stopped; if the convergence condition is not met, the next iteration is continued, and the above steps are repeated.

[0139] As can be clearly understood from the foregoing description, embodiments of the present disclosure propose a wave function optimization method suitable for quantum systems. The method realizes the neural scaling law between the wave function optimization accuracy and the neural network size based on the LAVA combined with the variational and projection framework. Specifically, embodiments of the present disclosure follow a clear power law relationship with the increase of the neural network size and the computing resources, thereby gradually approaching the exact solution of the non-relativistic many-electron Schrödinger equation. In this way, it helps to solve a plurality of long-term problems in quantum chemistry. Specifically, embodiments of the present disclosure can realize the reduction of the cyclic ozone transition barrier to nearly 0.5 electron volts, and for the first time, it confirms through calculation that the cyclic ozone is unstable in the ozone layer at low temperature. In the nitrogen dimer and cyclobutadiene system, the energy level benchmarks generated by embodiments of the present disclosure are superior to the control data in both accuracy and reliability. Finally, embodiments of the present disclosure have wide effectiveness in transition metal and benzene scale organic molecules. These achievements not only verify the universality of the neural scaling law, but also provide reliable benchmarks for approaching the exact solution of the non-relativistic Schrödinger equation.

[0140] Embodiments of the present disclosure also provide a corresponding apparatus for implementing the above method or process.Figure 4 A schematic structural block diagram of an apparatus 400 for wave function optimization according to some embodiments of the present disclosure is shown. The apparatus 400 can be implemented as or included in the electronic device 120. Various modules / components in the apparatus 400 can be implemented by hardware, software, firmware, or any combination thereof.

[0141] With reference to Figure 4 The apparatus 400 includes a trial wave function representation determination module 410, a quantum state determination module 420, and a wave function representation update module 430. The trial wave function representation determination module 410 is configured to obtain, from a first wave function representation of an estimated quantum system of a reference quantum state, a first trial wave function representation by reducing an energy of the estimated quantum system of the reference quantum state in a parameter space corresponding to a wave function of the quantum system. The quantum state determination module 420 is configured to determine an estimate of the first reference quantum state by reducing an energy estimate corresponding to the first trial wave function representation of the reference quantum state in a physical space corresponding to the estimated quantum system of the reference quantum state. The wave function representation update module 430 is configured to update the first wave function representation of the reference quantum state to a second wave function representation in the estimated parameter space of the reference quantum state based on the estimated first reference quantum state.

[0142] In some embodiments, the quantum state determination module 420 is further configured to determine a combined wave function representation based on the first wave function representation and the first trial wave function representation, determine local energies of a plurality of first samples of the quantum system based on the combined wave function representation and an energy operator of the quantum system, and determine the first reference quantum state based on the local energies of the plurality of first samples.

[0143] In some embodiments, the quantum state determination module 420 is further configured to determine a weight for the first trial wave function representation based on an inner product of the first wave function representation and an inner product of the first trial wave function representation, and determine the combined wave function representation based on the first wave function representation and the first trial wave function representation adjusted by the weight.

[0144] In some embodiments, the first wave function representation corresponds to a first parameter in the parameter space, and the wave function representation update module 430 is further configured to determine a parameter gradient at the first parameter in the parameter space based on the first reference quantum state, and adjust the first parameter to a second parameter corresponding to the second wave function representation in the parameter space based on the parameter gradient.

[0145] In some embodiments, the wave function representation update module 430 is further configured to determine a first energy gradient for the first wave function representation based on at least a local energy of the quantum system corresponding to the first reference quantum state and a relative relationship between the local energy and the reference variable, and convert the first energy gradient to the parameter gradient.

[0146] In some embodiments, the wave function representation updating module 430 is further configured to determine a first energy gradient for the first wave function representation based on a local energy of the quantum system corresponding to the first reference quantum state, a relative relationship between the local energy and the reference variable, and a sensitivity of the first wave function representation to an adjustment magnitude of the parameter.

[0147] In some embodiments, at least one of the first wave function representation, the second wave function representation, and the first trial wave function is determined based on a target neural network, wherein a parameter quantity of the target neural network is determined by: determining local energy variances of a plurality of second samples of the quantum system based on wave function representations of a plurality of reference neural networks having different parameter quantities; fitting a dependent variable of the target neural network based on the local energy variances; and determining the parameter quantity of the target neural network based on the dependent variable and a target accuracy of the wave function optimization.

[0148] In some embodiments, the trial wave function representation determining module 410 is further configured to adjust the first wave function representation based on a Monte Carlo algorithm until an energy of the quantum system is reduced to satisfy an energy requirement, and determine the first trial wave function representation based on the adjusted first wave function representation.

[0149] In some embodiments, the trial wave function representation determining module 410 is further configured to determine local energies of a plurality of third samples in the quantum system based on the first wave function representation and an energy operator of the quantum system, determine a second energy gradient for the first wave function representation based on the local energies of the plurality of third samples and a total energy of the quantum system corresponding to the first wave function representation, and adjust the first wave function representation based on the second energy gradient adjusted by a curvature approximation via a Kronecker factorization to determine the first trial wave function representation.

[0150] In some embodiments, the apparatus 400 further comprises a physical property determining module. The trial wave function representation determining module 410 is further configured to obtain a second trial wave function representation from the second wave function representation by reducing an energy of the quantum system in the parameter space. The quantum state determining module 420 is further configured to determine an estimate of a second reference quantum state by reducing an energy estimate corresponding to the second trial wave function representation in the physical space. The wave function representation updating module 430 is further configured to update the second wave function representation to a third wave function representation in the parameter space based on the second reference quantum state. The physical property determining module is configured to determine a physical property of the quantum system based on the third wave function representation in response to the third wave function representation satisfying a convergence condition.

[0151] Figure 5A block diagram of an electronic device 500 in which one or more embodiments of the disclosure can be implemented is shown. The electronic device 500 may, for example, be used to implement the electronic device 120 shown in FIG. 1, the apparatus 400 shown in FIG. 2, or the electronic device 300 shown in FIG. 3. It should be understood that the electronic device 500 shown is merely an example which should not limit the scope of functionality or Figure 1 Figure 4 Figure 5 The electronic device 500 shown is merely an example which should not limit the scope of functionality or

[0152] Referring to Figure 5 , the electronic device 500 is in the form of a general-purpose electronic device. Components of the electronic device 500 can include, but are not limited to, one or more processors 510, a memory 520, a storage device 530, one or more communication units 540, one or more input devices 550, and one or more output devices 560. The processor 510 can be a real or virtual processor and is capable of performing various processing according to programs stored in the memory 520. In a multi-processor system, multiple processors perform computer-executable instructions in parallel to improve parallel processing capability of the electronic device 500.

[0153] The electronic device 500 typically includes a number of computer storage media. Such media can be any available media that is accessible by the electronic device 500 and includes both volatile and non-volatile media, removable and non-removable media. The memory 520 can be a volatile memory (e.g., registers, cache, random access memory (RAM)), a non-volatile memory (e.g., read-only memory (ROM), electrically programmable read only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), flash memory), or some combination thereof. The storage device 530 can be a removable or non-removable media and can include a machine-readable medium, such as a flash drive, a magnetic disk, or any other medium that can be used to store information and / or data and that can be accessed by the electronic device 500.

[0154] The electronic device 500 can further include additional removable / non-removable, volatile / non-volatile storage media. Although not shown in Figure 5 , a disk drive for reading from or writing to a removable, non-volatile magnetic disk (e.g., a "floppy disk"), and an optical disk drive for reading from or writing to a removable, non-volatile optical disk (e.g., a CD-ROM) can be provided. In these instances, each drive can be connected to the bus (not shown) by one or more data media interfaces. The memory 520 can include a computer program product 525 having one or more program modules configured to carry out the various methods or actions of the various embodiments of the present disclosure.

[0155] ​​The communication unit 540 enables communication through the communication medium with other electronic devices. Additionally, the functionality of the components of the electronic device 500 can be implemented in a single computing cluster or a plurality of computer machines that are capable of communicating over a communication connection. As such, the electronic device 500 can operate in a networked environment using logical connections to one or more other servers, network personal computers (PCs), or another network node.

[0156] The input device 550 can be one or more input devices, such as a mouse, a keyboard, a trackball, etc. The output device 560 can be one or more output devices, such as a display, a speaker, a printer, etc. The electronic device 500 can also communicate with one or more external devices (not shown), such as a storage device, a display device, etc., through the communication unit 540, as necessary, with one or more devices that enable a user to interact with the electronic device 500, or with any device (e.g., a network card, a modem, etc.) that enables the electronic device 500 to communicate with one or more other electronic devices. Such communication can be carried out via an input / output (I / O) interface (not shown).

[0157] According to an example implementation of the present disclosure, there is provided a computer-readable storage medium having computer-executable instructions stored thereon, where the computer-executable instructions are executed by a processor to implement the method described above. According to an example implementation of the present disclosure, there is also provided a computer program product tangibly stored on a non-transitory computer-readable medium and including computer-executable instructions, where the computer-executable instructions are executed by a processor to implement the method described above.

[0158] Various aspects of the disclosure are now described with reference to the drawings. In general, the drawings described below are diagrammatic and schematic representations of actual or conceptual structures and processes, and are not limiting of the scope of the present disclosure. In the drawings, the same reference numerals are used to represent similar or like items.

[0159] These computer readable program instructions can be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks. These computer readable program instructions can also be stored in a computer readable storage medium that can include a non-transitory computer readable medium, thereby making a computer program product, i.e., an article of manufacture, including instructions which implement various aspects of the functions / acts specified in the flowchart and / or block diagram block or blocks. The computer readable medium can be a machine readable storage medium or a machine readable transmission medium. Furthermore, the one or more computer readable storage media mentioned above could be implemented with a single physical element.

[0160] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable data processing apparatus or other device to produce a computer implemented process such that the instructions which execute on the computer, other programmable data processing apparatus, or other device implement the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0161] The flow diagrams and the block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various implementations of the present disclosure. In this regard, each block in the flow diagrams and the block diagrams can represent a module, segment, or portion of instructions, which comprises one or more executable instructions for implementing the specified logic functions (s). In some alternative implementations, the functions noted in the blocks can occur out of the order noted in the figures. For example, two blocks shown in succession may, in some cases, be executed substantially concurrently, or the blocks can sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and / or flowchart illustrations, and combinations thereof, can be implemented by a dedicated hardware-based system that performs the specified functions or acts, or combinations of hardware and software.

[0162] implementations. Numerous modifications and adaptations will be apparent to those skilled in the art without departing from the scope and spirit of the described implementations. Specific details known to those of ordinary skill in the art are included for purposes of providing a complete description of the implementations, but are not intended to limit the scope of the described implementations. Determination of the scope of the described implementations is meant to be made by the appended claims and their equivalents.

Claims

1. A wave function optimization method, comprising: In a parameter space corresponding to a wave function of a quantum system, obtaining a first trial wave function representation from a first wave function representation of the quantum system by reducing the energy of the quantum system; determining an estimate of a first reference quantum state by reducing an energy estimate corresponding to the first trial wave function representation in a physical space corresponding to the quantum system; and Based on the first reference quantum state, the first wave function representation is updated to a second wave function representation in the parameter space.

2. The method of claim 1 , wherein determining an estimate of a first reference quantum state by reducing an energy estimate corresponding to the first trial wave function representation comprises: determining a combined wave function representation based on the first wave function representation and the first trial wave function representation; determining local energies of a plurality of first samples of the quantum system based on the combined wave function representation and an energy operator of the quantum system; and An estimate of the first reference quantum state is determined based on the local energies of the plurality of first samples.

3. The method of claim 2 , wherein determining the combined wave function representation comprises: determining a weight for the first trial wave function representation based on an inner product of the first wave function representation and an inner product of the first trial wave function representation; as well as The combined wave function representation is determined based on the first wave function representation and the first tentative wave function representation after the weight adjustment.

4. The method of claim 1 , wherein the first wave function representation corresponds to a first parameter in the parameter space, and updating the first wave function representation to a second wave function representation in the parameter space comprises: determining a parameter gradient at the first parameter in the parameter space based on the first reference quantum state; as well as In the parameter space, based on the parameter gradient, the first parameter is adjusted to a second parameter corresponding to the second wave function representation.

5. The method of claim 4, wherein determining the parameter gradient comprises: determining a first energy gradient for the first wave function representation based at least on a local energy of the quantum system corresponding to the first reference quantum state and a relative relationship between the local energy and a reference variable; and The first energy gradient is converted into the parameter gradient.

6. The method of claim 5 , wherein determining a first energy gradient for the first wave function representation based on at least a local energy of the quantum system corresponding to the first reference quantum state and a relative relationship between the local energy and a reference variable comprises: The first energy gradient for the first wave function representation is determined based on the local energy, the relative relationship between the local energy and the reference variable, and the sensitivity of the first wave function representation to the adjustment amplitude.

7. The method of claim 1 , wherein at least one of the first wave function representation, the second wave function representation, and the first trial wave function is determined based on a target neural network, wherein the parameter amount of the target neural network is determined by: determining local energy variances of a plurality of second samples of the quantum system based on wave function representations of a plurality of reference neural network outputs having different parameter amounts; Fitting the dependent variables of the target neural network based on the local energy variance; and Based on the dependent variables and the target accuracy of wave function optimization, the parameter quantity of the target neural network is determined.

8. The method of claim 1 , wherein obtaining a first trial wave function representation from a first wave function representation of the quantum system by reducing the energy of the quantum system comprises: Adjusting the first wave function representation based on a Monte Carlo algorithm until the energy of the quantum system is reduced to meet energy requirements; as well as Based on the adjusted first wave function representation, the first tentative wave function representation is determined.

9. The method of claim 8, wherein adjusting the first wave function representation based on a Monte Carlo algorithm comprises: determining local energies of a plurality of third samples in the quantum system based on the first wave function representation and an energy operator of the quantum system; determining a second energy gradient for the first wave function representation based on the local energies of the plurality of third samples and a total energy of the quantum system corresponding to the first wave function representation; and The first wave function representation is adjusted based on the second energy gradient after the Kronecker factored approximate curvature adjustment to determine the first tentative wave function representation.

10. The method according to claim 1, further comprising: In the parameter space, obtaining a second trial wave function representation from the second wave function representation by reducing the energy of the quantum system; determining, in the physical space, an estimate of a second reference quantum state by reducing an estimate of an energy corresponding to the second trial wave function representation; Based on the second reference quantum state, updating the second wave function representation to a third wave function representation in the parameter space; as well as In response to the third wave function representation satisfying a convergence condition, a physical property of the quantum system is determined based on the third wave function representation.

11. A device for wave function optimization, comprising: a trial wave function representation determination module configured to obtain a first trial wave function representation from a first wave function representation of the quantum system by reducing the energy of the quantum system in a parameter space corresponding to the wave function of the quantum system; a quantum state determination module configured to determine an estimate of a first reference quantum state by reducing an energy estimate corresponding to the first trial wave function representation in a physical space corresponding to the quantum system; and A wave function representation updating module is configured to update the first wave function representation to a second wave function representation in the parameter space based on the first reference quantum state.

12. An electronic device comprising: at least one processor; as well as At least one memory is coupled to the at least one processor and stores instructions for execution by the at least one processor, the instructions causing the electronic device to perform the method according to any one of claims 1 to 10 when executed by the at least one processor.

13. A computer-readable storage medium having computer-executable instructions stored thereon, wherein the computer-executable instructions can be executed by a processor to implement the method according to any one of claims 1 to 10.

14. A computer program product comprising computer executable instructions, wherein the computer executable instructions, when executed by a processor, implement the method according to any one of claims 1 to 10.

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