Method, apparatus, device and storage medium for determining quantum system topology type

Abstract

A method, apparatus, device, and storage medium for determining the topological type of a quantum system are provided. The proposed method includes: using a neural network for the quantum system, determining multiple quantum states of the quantum system by inserting magnetic flux into the quantum system, wherein the neural network represents the wave function of the quantum system, and the multiple quantum states correspond to the inserted multiple magnetic fluxes; determining the corresponding polarization states of the quantum system under the multiple quantum states based on the neural network; and determining the topological type of the quantum system based on the corresponding polarization states under the multiple quantum states. In this manner, by inserting magnetic flux into the quantum system and utilizing a neural network, the topological type of the system can be determined, thereby facilitating the acquisition of more properties of the system.

Description

Technical Field

[0001] The examples in this paper generally relate to the field of computer science, and in particular to methods, apparatuses, devices, and computer-readable storage media for determining the topological type of a quantum system. Background Technology

[0002] Correlated topological states (such as fractional Chern insulators) represent a cutting-edge area of ​​research in condensed matter physics. These quantum states exhibit exotic quantum phenomena such as fractionally quantized Hall conductance and anyon statistics. In recent years, the study of correlated topological states in molar systems such as torsional bilayer graphene and transition metal dichalcogenides has received widespread attention. Summary of the Invention

[0003] In a first aspect, a method for determining the topological type of a quantum system is provided. The method includes: using a neural network for the quantum system, determining multiple quantum states of the quantum system by inserting magnetic flux into the quantum system, the neural network representing the wave function of the quantum system, the multiple quantum states corresponding to the inserted multiple magnetic fluxes; determining corresponding polarization states of the quantum system under the multiple quantum states based on the neural network; and determining the topological type of the quantum system based on the corresponding polarization states under the multiple quantum states.

[0004] In a second aspect, an apparatus for determining the topology type of a quantum system is provided. The apparatus includes: a quantum state determination module configured to determine multiple quantum states of the quantum system by inserting magnetic flux into the quantum system using a neural network representing the wave function of the quantum system, the multiple quantum states corresponding to the inserted multiple magnetic fluxes; a polarization state determination module configured to determine corresponding polarization states of the quantum system under the multiple quantum states based on the neural network; and a topology type determination module configured to determine the topology type of the quantum system based on the corresponding polarization states under the multiple quantum states.

[0005] In a third aspect, an electronic device is provided. The device includes 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. When executed by the at least one processor, the instructions cause the device to perform the method of the first aspect.

[0006] In a fourth aspect, a computer-readable storage medium is provided. The computer-readable storage medium stores computer-executable instructions that can be executed by a processor to implement the method of the first aspect.

[0007] In a fifth aspect, a computer program product is provided, which is tangibly stored in a computer storage medium and includes computer-executable instructions that, when executed by a device, cause the device to perform the method of the first aspect.

[0008] In this way, by inserting the magnetic flux of a quantum system and utilizing neural networks, the topological type of the system can be determined, which is beneficial for obtaining more properties of the system.

[0009] It should be understood that the content described in this section is not intended to limit the key or important features of the examples in this article, nor is it intended to restrict the scope of the solution. Other features will become readily apparent from the following description. Attached Figure Description

[0010] The above and other features, advantages, and aspects of the various examples herein will become more apparent when taken in conjunction with the accompanying drawings and the following detailed description. In the accompanying drawings, the same or similar reference numerals denote the same or similar elements, wherein: Figure 1 The diagrams show example environments for some scenarios; Figure 2 A flowchart illustrating an example process for determining the topological type of a quantum system based on several scenarios is shown. Figure 3 A schematic diagram of a neural network structure for representing a multibody wave function is shown, depending on several scenarios. Figure 4 A schematic diagram illustrating the relationship between polarization, magnetic flux insertion, and charge pumping in parameter space under certain conditions is shown. Figure 5A A schematic diagram of the toroidal geometry of a trivial insulator under unit magnetic flux insertion is shown, according to some cases. Figure 5B A schematic diagram of the toroidal geometry of a Chen insulator under unit magnetic flux insertion is shown, according to some cases. Figure 5C A schematic diagram of the toroidal geometry of a fractional-Chen insulator under multiple unit magnetic flux insertions is shown, according to some cases. Figure 6 The diagram shows graphs for determining the topology type of a quantum system under several conditions; Figure 7 A schematic block diagram of a device for determining the topology type of a quantum system under certain conditions is shown. Figure 8 A block diagram of an electronic device in which some scenarios can be implemented is shown. Detailed Implementation

[0011] The examples in the text will now be described in more detail with reference to the accompanying drawings. While some examples are shown in the drawings, it should be understood that solutions can be implemented in various forms and should not be construed as limited to the examples presented herein. Rather, these examples are provided to provide a more thorough and complete understanding of the solutions. It should be understood that the drawings and examples in this document are for illustrative purposes only and are not intended to limit the scope of protection of the solutions.

[0012] It should be noted that the headings of any section / subsection provided herein are not restrictive. Various examples are described throughout this document, and examples of any type may be included under any section / subsection. Furthermore, examples described in any section / subsection may be combined in any way with any other examples described in the same section / subsection and / or different sections / subsections.

[0013] In the description of the examples in this document, the term "including" and similar terms should be understood as open inclusion, i.e., "including but not limited to". The term "based on" should be understood as "at least partially based on". The term "an example" or "the example" should be understood as "at least one example". The term "some examples" should be understood as "at least some examples". Other explicit and implicit definitions may also be included below. The terms "first", "second", etc., may refer to different or the same objects. Other explicit and implicit definitions may also be included below.

[0014] The examples in this document may involve user data, data acquisition, and / or use. All of these aspects comply with relevant laws, regulations, and provisions. In the examples presented herein, all data collection, acquisition, processing, manipulation, forwarding, and use are conducted with the user's knowledge and confirmation. Accordingly, when implementing each example, the type, scope of use, and usage scenarios of any data or information that may be involved should be communicated to the user and their authorization obtained through appropriate means, in accordance with relevant laws and regulations. The specific methods of notification and / or authorization can vary depending on the actual situation and application scenario; the scope of the solution is not limited in this regard.

[0015] In this manual and the sample solutions, any processing of personal information will be conducted only under legal grounds (such as obtaining the consent of the data subject or being necessary for the performance of a contract) and will only be carried out within the scope stipulated or agreed upon. A user's refusal to process personal information beyond what is necessary for basic functions will not affect the user's use of basic functions.

[0016] In this paper, "quantum system" can refer to any physical system with quantum mechanical properties, including but not limited to multi-electron systems, Mohr's superlattice systems, torsional bilayer graphene systems, and transition metal dichalcogenide systems. "Topological type" can refer to the classification of the topological properties of a quantum system, such as trivial insulators, Chern insulators, fractional Chern insulators, etc. "Topological invariants" can refer to quantities characterizing topological properties, such as Chern numbers (e.g., many-body Chern numbers), Berry connections, and Berry curvature. "Polarization state" can refer to quantities reflecting the overall shift in charge distribution within a quantum system, such as reflecting the position of the charge distribution center in a periodic system. "Magnetic flux insertion" can refer to simulating the process of magnetic flux passing through the system by changing the boundary conditions; this process can be equivalent to torturing the boundary conditions in this paper.

[0017] As mentioned above, correlated topological states (such as fractional Chern insulators (FCIs)) are at the forefront of condensed matter physics research, where exotic quantum phenomena such as fractionally quantized Hall conductance and anyon statistics can be realized. In recent years, the study of correlated topological states in molar systems such as torsional bilayer graphene and transition metal chalcogenides has received widespread attention. These systems possess flat and topologically nontrivial band structures, and the synergistic effect of strong many-body correlations and nontrivial quantum geometry can stabilize exotic quantum phases, including fractional Chern insulators and composite Fermi liquids under zero magnetic field.

[0018] From a theoretical perspective, correlated topological states pose challenges to theoretical description. When strong electron correlation is indispensable, a many-body description is needed that goes beyond the effective single-particle picture, and the accurate calculation of the many-body wavefunction itself has been a long-standing challenge. Neural network variational Monte Carlo methods, utilizing the expressive power of neural networks to represent many-body wavefunctions, have become a powerful approach to address this challenge. However, extracting reliable topological invariants from numerically obtained many-body wavefunctions remains difficult.

[0019] A scheme for determining the topology type of a quantum system is proposed. This scheme includes: using a neural network for the quantum system, determining multiple quantum states of the quantum system by inserting magnetic flux into the quantum system; the neural network representing the wave function of the quantum system, with each quantum state corresponding to one of the inserted magnetic fluxes; determining the corresponding polarization states of the quantum system under the multiple quantum states based on the neural network; and determining the topology type of the quantum system based on the corresponding polarization states under the multiple quantum states.

[0020] By employing the above-described technical solutions, the topological type of a quantum system can be determined based on a pumping mechanism (e.g., a Thouless pumping mechanism). For example, the system can be subjected to an adiabatic and periodic modulation process (e.g., magnetic flux insertion), during which the polarization response can be monitored to extract topological information, such as the many-body Chern number.

[0021] As will become clear from the following description, the proposed scheme is independent of the microscopic details of electron-electron interactions and is applicable to strongly interacting systems. This enables the reliable calculation of topological invariants in correlated topological systems, such as fractional Chern insulators, on a larger scale. Furthermore, the proposed scheme is applicable to topological states of different topological classifications, such as Abelian FCI states, non-Abelian FCI states, and composite Fermi liquid states, thus allowing for broad application to various topological insulator materials.

[0022] The following describes various examples of this scheme in further detail with reference to the accompanying drawings.

[0023] Example Environment Figure 1 A schematic diagram of example environment 100 is shown. (e.g.) Figure 1 As shown, example environment 100 may include electronic device 110.

[0024] In this example environment 100, user 130 can interact with electronic device 120 via terminal device 110 and / or its attached devices. As an example, user 130 can provide information related to the quantum system to be analyzed to electronic device 120 via terminal device 110. After identifying the quantum system to be analyzed, electronic device 120 can establish a wavefunction representation of the quantum system and continuously optimize the wavefunction representation to make the energy estimate corresponding to the wavefunction representation approximate the true ground state energy of the quantum system. After optimizing the ground state wavefunction representation, electronic device 120 can determine various physical properties of the quantum system based on the optimized ground state wavefunction representation and provide them to user 130 for further analysis and processing.

[0025] In some cases, a user can provide information about changes in system parameters (e.g., applied torsional boundary conditions) to an electronic device 120 via a terminal device 110. The electronic device 120 can then use this information to re- or continuously optimize the wavefunction based on wavefunction methods (e.g., neural network variational Monte Carlo) to obtain quantum state wavefunction representations under different parameter conditions. This can, for example, provide a basis for subsequent calculations of topological invariants.

[0026] In some cases, electronic device 110 communicates with server 130 to provide services to application 120. Electronic device 110 can be any type of mobile terminal, fixed terminal, or portable terminal, including mobile phones, desktop computers, laptop computers, notebook computers, netbook computers, tablet computers, media computers, multimedia tablets, handheld computers, portable gaming terminals, VR / AR devices, personal communication system (PCS) devices, personal navigation devices, personal digital assistants (PDAs), audio / video players, digital cameras / camcorders, positioning devices, television receivers, radio receivers, e-book devices, gaming devices, or any combination of the foregoing, including accessories and peripherals of these devices or any combination thereof. In some cases, electronic device 110 can also support any type of user-facing interface (such as "wearable" circuitry).

[0027] Server 130 can be a standalone physical server, a server cluster or distributed system composed of multiple physical servers, or a cloud server providing basic cloud computing services such as cloud services, cloud databases, cloud computing, cloud functions, cloud storage, network services, cloud communication, middleware services, domain name services, security services, content delivery networks, and big data and artificial intelligence platforms. Server 130 may include, for example, computing systems / servers such as mainframes, edge computing nodes, computing devices in a cloud environment, etc. Server 130 can provide backend services for applications 120 in electronic devices 110 that support determining the topology type of quantum systems.

[0028] A communication connection can be established between server 130 and electronic device 110. This communication connection can be established via wired or wireless means. The communication connection can include, but is not limited to, Bluetooth, mobile network, Universal Serial Bus (USB), and Wireless Fidelity (WiFi) connections. In some cases, server 130 and electronic device 110 can exchange signaling information through their communication connection.

[0029] It should be understood that the structure and function of the various elements in environment 100 are described for illustrative purposes only and do not imply any limitation on the scope of the scheme.

[0030] The following description of the example will continue with reference to the accompanying drawings.

[0031] Figure 2A flowchart of a method 200 for determining the topological type of a quantum system is shown. Method 200 can be derived from... Figure 1 The electronic device 120 in the middle performs the operation.

[0032] In box 210, electronic device 120 uses a neural network for the quantum system to determine multiple quantum states of the quantum system by inserting magnetic flux into the quantum system. The neural network represents the wave function of the quantum system, and the multiple quantum states correspond to the multiple inserted magnetic fluxes.

[0033] In some cases, neural networks can employ a first-principles quantization scheme to directly map the electronic coordinates in continuous real space to the corresponding many-body wavefunction values. Figure 3 An example structure 300 for representing a neural network for multibody wave functions is shown. For example... Figure 3 As shown, the neural network maps electronic coordinates to the wave function value Ψ. The network's input layer receives multiple electronic coordinates, including the first electronic coordinate r1, the second electronic coordinate r2, and subsequent electronic coordinates up to the Nth electronic coordinate rN. Nodes in the input layer are connected to nodes in the intermediate hidden layers via connection lines. The network contains multiple hidden layers, which are interconnected through fully connected layers. Finally, the outputs of the hidden layers converge at the output terminal, generating the wave function value Ψ.

[0034] In some cases, magnetic flux insertion can be achieved by altering the boundary conditions of the quantum system. For systems with periodic boundary conditions, magnetic flux insertion is equivalent to torturing the boundary conditions. Torturing the boundary conditions is achieved by altering the periodic boundary conditions (e.g., which can be expressed as...). Based on the previous method, it is obtained by introducing phase twist (also called boundary phase or phase twist angle) when the wave function crosses the system boundary. For example, it can be obtained by... and Indicates phase torsion, with a value of Applying a unit magnetic flux to the system is numerically equivalent to applying a 2... The boundary phase twist. Based on this equivalence relation, the phase twist angle (i.e., and The term "magnetic flux insertion" indicates the insertion of magnetic flux. In the following text, the term "magnetic flux insertion" may be referred to as the phase twist angle (including but not limited to phase twist, boundary phase, etc.) and vice versa.

[0035] In some cases, topological invariants can be expressed as: (1) in Represents the multibody wave function. and As described above, this represents magnetic flux insertion (equivalent to phase twist angle). The many-body Chern number is defined as the integral over the entire space of the difference of the inner products of the geometric partial derivatives of the many-body wavefunctions with respect to the magnetic flux insertion, expressed through a factorial. Normalize.

[0036] Now, example scenarios for determining multiple quantum states of a quantum system are described. In some example scenarios, electronic device 120 can train a neural network based on a quantum system without magnetic flux insertion to obtain an initial quantum state as one of multiple quantum states, the initial quantum state corresponding to the case where the magnetic flux is zero. Electronic device 120 can further incrementally update the neural network from the initial quantum state by incrementally inserting magnetic flux into the quantum system to obtain the remaining quantum states other than the initial quantum state among the multiple quantum states. Specific details of inserting magnetic flux will be combined below. Figure 4 Describe it.

[0037] In some cases, the electronic device 120 can first acquire an initial quantum state using a variational optimization neural network under no-twist conditions (i.e., no magnetic flux insertion). Then, by changing the phase torsion angle (i.e., changing the magnetic flux insertion) and using the neural network wavefunction from the smaller torsion angle as the initial state, training continues until convergence. In this way, a series of quantum states corresponding to different magnetic fluxes can be obtained.

[0038] The process of incrementally updating the neural network is now described. For one of the multiple increments, based on the magnetic flux inserted into the quantum system in that increment, the electronic device 120 can determine the Hamiltonian for the quantum system. Based on the Hamiltonian and the neural network, the electronic device 120 can determine the energy of the quantum system. By using a Monte Carlo algorithm to reduce the energy of the quantum system, the electronic device 120 can update the neural network.

[0039] In some cases, Hamiltonian It can include non-interacting and interacting components. The non-interacting component describes the motion of a single electron in a periodic potential field, while the interacting component describes the Coulomb interaction between electrons. Magnetic flux insertion can be achieved by introducing a phase factor into the Hamiltonian. For example, the phase twist angle can be expressed as... and In the case of Hamiltonian The ground state can be represented as By applying a normalization transformation, It can represent the Hamiltonian without phase twisting. The ground state, where the phase factor .

[0040] In some cases, the electronic device 120 can use variational Monte Carlo methods to optimize the neural network parameters. Specifically, the expected energy can be estimated through importance sampling, and then the neural network parameters can be updated through gradient descent to reduce the energy.

[0041] In some cases, the polarization phase twist corresponding to the increasing magnetic flux inserted into the quantum system can increase from zero to integer multiples of a predetermined angle. For example, the phase twist angle can increase from 0 to... Or increment from 0 to (where q is an integer associated with the fractional fill factor). For a fractional Chern insulator, the ground state only exists during insertion. The magnetic flux will return to its original form only after it has passed through a certain point, during which time the polarization phase undergoes a certain process. The total displacement.

[0042] The parameters for training a neural network are now described. In some cases, the difference between training a neural network and incrementally updating it can include at least one of the following: training step size and learning rate. For example, the learning rate for initial training can be set to a large value (e.g., 3e-3), while the learning rate during charge pumping can be set to a smaller value (e.g., 3e-4). The number of training steps for initial training can be set to a large value (e.g., 4e4 steps), while the number of training steps during charge pumping can be set to a smaller value (e.g., 2e4 steps). This is because during charge pumping, the neural network wavefunction from a smaller torsion angle already provides a good initial state, thus requiring fewer training steps and a smaller learning rate to achieve convergence. Table 1 below provides examples of hyperparameters used in variational Monte Carlo sampling of neural networks, including settings for both neural networks and quantum neural networks. In the table, parameters related to initial training are used to calculate the initial state without phase torsion, and parameters related to pumping are used to calculate the charge pumping process achieved by changing the phase torsion.

[0043] Table 1. Hyperparameters used in the calculation

[0044] In box 220, electronic device 120 uses a neural network to determine the corresponding polarization states of the quantum system in multiple quantum states.

[0045] The process of determining the corresponding polarization state of a quantum system in multiple quantum states is now described. For each of the multiple quantum states, the electronic device 120 can determine the corresponding positions of multiple electrons in the quantum system in that quantum state based on a neural network corresponding to that quantum state. The electronic system 120 can determine the polarization state of the quantum system in that quantum state based on the wave vector in that quantum state and the corresponding positions of the multiple electrons.

[0046] In some cases, electronic system 120 can use the Resta polarization operator to determine the polarization state. The Resta polarization operator can be expressed as: Where b is the wave vector in the first Brillouin zone, It represents the position of the i-th electron. The phase of the expected value of the polarization operator can be interpreted as the periodic position of the electron.

[0047] In some cases, the electronic device 120 can sample polarization by fixing the parameters of a neural network and performing inference operations. Specifically, importance sampling can be used. To sample polarization The expected value.

[0048] Figure 4 A schematic diagram illustrating the relationship between polarization, magnetic flux insertion, and charge pumping in parameter space 400 is shown. Figure 4 As shown, parameter space 400 is represented as a rectangular area with a gray border. A large arrow labeled "Polarization" points to the left of parameter space 400, indicating the direction of polarization input. Within parameter space 400, an upward arrow labeled "Fluorescence Insertion" indicates the direction in which magnetic flux is inserted into the system. A horizontal double-headed arrow labeled "arg Z" represents the argument of the polarization operator within parameter space 400. Charge carrier 410 is depicted as a circular element located in the lower right portion of parameter space 400, with an arrow extending to the right labeled "Charge Pump," indicating the direction of charge transfer. Charge carrier 410 represents charge movement in response to the magnetic flux insertion process.

[0049] In some cases, the direction of the inserted magnetic flux may not be parallel to the polarization direction of the corresponding polarization state. For example, magnetic flux can be inserted adiabatically along the x-direction while monitoring polarization changes along the y-direction. This configuration allows topological information to be extracted via charge pumping processes.

[0050] In box 230, electronic device 120 determines the topology type of the quantum system based on the corresponding polarization states under multiple quantum states.

[0051] The process of determining the topological type of a quantum system is now described. In some cases, electronic system 120 can determine the polarization characteristics of the quantum system based on the corresponding polarization states in multiple quantum states. The polarization characteristics indicate the tendency of the quantum system's polarization to change with the inserted magnetic flux. Electronic system 120 can determine the topological type of the quantum system based on the polarization characteristics.

[0052] In some cases, polarization characteristics can be determined by plotting the polarization phase as a function of magnetic flux. The phase twist diagram can be approximated as linear, and the slope is the desired singlet Chern number. Specific examples of such plots will be provided below. Figure 6 To describe.

[0053] We now describe determining the topological type of a quantum system based on polarization properties. Electronic device 120 can determine the topological invariants of a quantum system based on its polarization properties. These topological invariants characterize the topological type.

[0054] In some cases, such as in Thouless pumps, the topological invariants represented by equation (1) can be calculated using the following formula: (2) Among them | It is the non-degenerate wave function of the momentum sector K. For the polarization operator , Indicates the initial state (twisted phase) Electrodeization under ) Indicates the twisted phase The polarization under the given conditions. The expected values ​​of the polarization operators for the two states are taken in phase, and the difference between the phases is the topological invariant in the charge pumping process.

[0055] Figure 5A , Figure 5B and Figure 5C A schematic diagram of charge pumping for systems with different topological orders is shown. Figure 5A The toroidal geometry of a trivial insulator under unit magnetic flux insertion is shown. The toroid is displayed in a three-dimensional perspective view, with a grid pattern on its surface representing the periodic boundary conditions of the system. An upward arrow 505, labeled "Unit Magnetic Flux," passes perpendicularly through the central hole of the toroid, indicating the direction of magnetic flux insertion along one axis. For trivial insulators, such as... Figure 5A As shown, after inserting a unit magnetic flux 505, the polarization phase changes only locally, and the trajectory 510 does not circumvent the torus, meaning that the flux insertion does not lead to charge transfer, and the Chern number C=0. The topological invariants correspond to the trajectory's circumvention, and thus to the topology type.

[0056] Figure 5B The toroidal geometry of a Chern insulator under unit magnetic flux insertion is shown. Figure 5A Similarly, the torus is shown in a three-dimensional perspective view, with a unit magnetic flux of 515 perpendicularly passing through the central hole. For Chen insulators, such as... Figure 5B As shown, trajectory 520 encircles the torus once, and the insertion of a unit magnetic flux 515 results in a pump of electron charge, with a Chern number C=1. That is, a Chern number C=1 indicates that the system is a Chern insulator.

[0057] Figure 5CThe toroidal geometry of a fractional Chen insulator under multiple unit magnetic flux insertions is shown. Multiple upward arrows 525 pass through the central hole of the toroid, indicating that p unit magnetic fluxes 525 are inserted into the system. The trajectories 530 on the toroidal surface indicate the evolution path of charge pump polarization during flux insertion, and the number of trajectories reflects the topological pump charge of the system. For a fractional Chen insulator, adiabatic magnetic flux insertion in one direction induces charge pumping in the perpendicular direction, and the total pump charge is related to the topological invariants of the system. Figure 5C In the example shown, the Chern number obtained by calculating the charge pump caused by a unit magnetic flux is a fraction (i.e., the number of revolutions 2 divided by the number of unit magnetic fluxes p), indicating that the system is a fractional Chern insulator.

[0058] Figure 6 A graph is shown to determine the topological type of a quantum system. Specifically, Figure 6 This is a graph showing the imaginary part of the natural logarithm of the expectation value of the polarization operator as a function of the magnetic flux ratio. The horizontal axis represents the ratio of the magnetic flux to the reference magnetic flux value, i.e., the ratio of the magnetic flux insertion to a unit magnetic flux. For example... Figure 6 As shown, in this example, the ratio ranges from 0.0 to 1.0. The vertical axis represents the imaginary part of the natural logarithm of the polarization expectation, expressed in terms of 2... The unit of measurement is 0.0, and in this example, it ranges from approximately -0.6. The imaginary part of the natural logarithm of the polarization operator's expectation value represents the phase of the polarization operator's expectation value, which is related to... Figures 5A-5C The trajectory on the torus shown corresponds to this. As described above, when a unit magnetic flux is inserted, the phase change of the expected value of the polarization operator (i.e., Figures 5A-5C The trajectory shown corresponds to a topological invariant (i.e., a Chern number). Therefore, in the case of... Figure 6 In the diagram shown, the slope of the curve can represent the topological invariant of the corresponding system, and thus the topological type of the system can be determined based on the slope.

[0059] Figure 6 Two data series derived from the neural network are plotted. The first data series 601, represented by a solid line with a solid circle, is labeled CDW (charge density wave), showing that the value remains near 0 throughout the magnetic flux range with a slope of 0.008. The second data series 602, represented by a dotted line with a solid circle, is labeled FCI (fractional Chern insulator), showing that the value linearly decreases from around 0 when the magnetic flux is zero to approximately -0.344 when the flux ratio is 1.0. A complex competition exists between the FCI and CDW states; subtle differences in computational settings can cause the trained neural network wavefunction to converge to completely different states, making the distinction between these two competing states extremely challenging. However, using the proposed method 200, these two quantum states can be precisely distinguished.

[0060] like Figure 6 As shown, the slope of the polarization curve of the CDW state obtained through the neural network is almost zero. This indicates that the Chern number C of the system is almost zero, meaning the system's topological type is topologically trivial, corresponding to the characteristics of the CDW state. Figure 6 The slope of the polarization curve for the FCI state, derived from the neural network, shown in the figure is approximately 1 / 3. This indicates that the Chern number of the system is approximately 1 / 3, corresponding to the one-third filling fraction of the FCI state. Therefore, this curve indicates that the system's topological type is a fractional Chern insulator. The comparison between the two data series demonstrates the different polarization responses of different quantum states during magnetic flux insertion, thus distinguishing between topologically nontrivial and topologically trivial states.

[0061] Figure 6 The proposed method demonstrates its universality, applicable not only to fractional Chern insulators but also to trivial insulators. Furthermore, it can be extended to topological states of different topological classifications, such as calculating topological invariants in Abelian FCI states, non-Abelian FCI states, and composite Fermi liquid states, thus enabling its wide application to various topological insulator materials. Additionally, although this disclosure describes the many-body case with a fixed total momentum sector (e.g., ...), ... Figure 6 The proposed scheme is applicable to the case of FCI states described in the previous paper, but it can also be applied to the more general quantum state cases corresponding to arbitrary beams in the Brillouin zone.

[0062] Using the method described above 200, topological invariants of quantum systems can be determined based on charge-pumping process simulations. This method utilizes the concept of Thouless pumping, where adiabatic and periodic modulation of system parameters returns the system to its initial state, with each cycle generating quantized charge transport associated with topological invariants. Topological information is extracted by monitoring the polarization response upon magnetic flux insertion. This method is independent of the microscopic details of electron-electron interactions and is applicable to strongly interacting systems. It reliably calculates topological invariants of correlated topological systems such as fractional Chern insulators in larger-scale systems and is stable across various topological classifications, providing a universal, stable, and accurate method for determining the physical properties of different topological insulator materials.

[0063] A corresponding apparatus for implementing the above methods or processes is also provided.

[0064] Figure 7 A block diagram of a topology type determination device 700 for several scenarios is shown. Device 700 can be implemented as or included in a server, terminal device, or cloud computing platform. The various modules / components in device 700 can be implemented by hardware, software, firmware, or any combination thereof.

[0065] like Figure 7 As shown, the device 700 includes a quantum state determination module 710, configured to determine multiple quantum states of the quantum system by inserting magnetic flux into the quantum system using a neural network for the quantum system. The neural network represents the wave function of the quantum system, and the multiple quantum states correspond to the multiple inserted magnetic fluxes, respectively. A polarization state determination module 720 is configured to determine the corresponding polarization states of the quantum system under the multiple quantum states based on the neural network. A topology type determination module 730 is configured to determine the topology type of the quantum system based on the corresponding polarization states under the multiple quantum states.

[0066] In some cases, the quantum state determination module 710 can be further configured to: train a neural network based on a quantum system without magnetic flux insertion to obtain an initial quantum state as one of a plurality of quantum states, the initial quantum state corresponding to a magnetic flux of zero; and incrementally update the neural network from the initial quantum state by incrementally inserting magnetic flux into the quantum system to obtain the remaining quantum states other than the initial quantum state among the plurality of quantum states.

[0067] In some cases, the quantum state determination module 710 can be further configured to: determine the Hamiltonian for the quantum system based on the magnetic flux inserted into the quantum system in one of the multiple increments; determine the energy of the quantum system based on the Hamiltonian and the neural network; and update the neural network by reducing the energy of the quantum system using a Monte Carlo algorithm.

[0068] In some cases, the polarization phase twist corresponding to the increasing magnetic flux inserted into the quantum system can increase from zero to an integer multiple of a predetermined angle.

[0069] In some cases, the difference between training a neural network and incrementally updating the neural network can include at least one of the following: training step size and learning rate.

[0070] In some cases, the polarization state determination module 720 can be further configured to: for each of the multiple quantum states, determine the corresponding positions of multiple electrons in the quantum system under that quantum state based on the neural network corresponding to that quantum state; and determine the polarization state of the quantum system under that quantum state based on the wave vector under that quantum state and the corresponding positions of the multiple electrons.

[0071] In some cases, the direction of the inserted magnetic flux may not be parallel to the polarization direction of the corresponding polarization state.

[0072] In some cases, the topology type determination module 730 can be further configured to: determine the polarization characteristics of the quantum system based on the corresponding polarization states under multiple quantum states, the polarization characteristics indicating the trend of the polarization of the quantum system as a function of the inserted magnetic flux; and determine the topology type of the quantum system based on the polarization characteristics.

[0073] In some cases, the topology type determination module 730 can be further configured to: determine the topological invariants of the quantum system based on polarization properties, wherein the topological invariants characterize the topology type.

[0074] The modules included in device 700 can be implemented in various ways, including software, hardware, firmware, or any combination thereof. In some cases, one or more modules can be implemented using software and / or firmware, such as machine-executable instructions stored on a storage medium. In addition to or as an alternative to machine-executable instructions, some or all of the units in device 700 can be implemented at least partially by one or more hardware logic components. By way of example, and not limitation, exemplary types of hardware logic components that can be used include field-programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), application-specific standard parts (ASSPs), systems on a chip (SOCs), complex programmable logic devices (CPLDs), and so on.

[0075] Figure 8 A block diagram of an electronic device 800 in which one or more examples may be implemented is shown. It should be understood that... Figure 8 The electronic device 800 shown is merely exemplary and should not be construed as limiting the functionality and scope of the examples described herein. Figure 8 The electronic device 800 shown can be used to implement the topology type determination device 700 as discussed above.

[0076] like Figure 8As shown, electronic device 800 is in the form of a general-purpose electronic device. Components of electronic device 800 may include, but are not limited to, one or more processing units or processors 810, memory 820, storage devices 830, one or more communication units 840, one or more input devices 850, and one or more output devices 860. Processor 810 may be a physical or virtual processor and is capable of performing various processes according to programs stored in memory 820. In a multiprocessor system, multiple processors execute computer-executable instructions in parallel to improve the parallel processing capability of electronic device 800.

[0077] Electronic device 800 typically includes multiple computer storage media. Such media can be any accessible media that is accessible to electronic device 800, including but not limited to volatile and non-volatile media, removable and non-removable media. Memory 820 can be volatile memory (e.g., registers, cache, random access memory (RAM)), non-volatile memory (e.g., read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory), or some combination thereof). Storage device 830 can be removable or non-removable media and can include machine-readable media, such as flash drives, disks, or any other media that can be used to store information and / or data and can be accessed within electronic device 800.

[0078] Electronic device 800 may further include additional removable / non-removable, volatile / non-volatile storage media. Although not explicitly stated... Figure 8 As shown, disk drives for reading from or writing to removable, non-volatile disks (e.g., "floppy disks") and optical disk drives for reading from or writing to removable, non-volatile optical disks can be provided. In these cases, each drive can be connected to a bus (not shown) via one or more data media interfaces. Memory 820 may include computer program product 825 having one or more program modules configured to perform various methods or actions of various examples.

[0079] The communication unit 840 enables communication with other electronic devices via a communication medium. Additionally, the functionality of the components of the electronic device 800 can be implemented using a single computing cluster or multiple computing machines capable of communicating via communication connections. Therefore, the electronic device 800 can operate in a networked environment using logical connections to one or more other servers, networked personal computers, or another network node.

[0080] Input device 850 can be one or more input devices, such as a mouse, keyboard, trackball, etc. Output device 860 can be one or more output devices, such as a monitor, speaker, printer, etc. Electronic device 800 can also communicate with one or more external devices (not shown) via communication unit 840 as needed. External devices include storage devices, display devices, etc., and can communicate with one or more devices that enable user interaction with electronic device 800, or with any device that enables electronic device 800 to communicate with one or more other electronic devices (e.g., network card, modem, etc.). Such communication can be performed via input / output (I / O) interface (not shown).

[0081] A computer-readable storage medium is provided that stores computer-executable instructions thereon, wherein the computer-executable instructions are executed by a processor to implement the methods described above. A computer program product is also provided, which is tangibly stored on a non-transitory computer-readable medium and includes computer-executable instructions, which are executed by a processor to implement the methods described above.

[0082] The flowcharts and / or block diagrams of the methods, apparatus, devices, and computer program products referred to herein describe various aspects. It should be understood that each block of the flowcharts and / or block diagrams, as well as combinations of blocks in the flowcharts and / or block diagrams, can be implemented by computer-readable program instructions.

[0083] These computer-readable program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing apparatus to produce a machine such that, when executed by the processor of the computer or other programmable data processing apparatus, they create means for implementing the functions / actions specified in one or more blocks of the flowchart and / or block diagram. These computer-readable program instructions can also be stored in a computer-readable storage medium that causes a computer, programmable data processing apparatus, and / or other device to operate in a particular manner; thus, the computer-readable medium storing the instructions comprises an article of manufacture that includes instructions for implementing aspects of the functions / actions specified in one or more blocks of the flowchart and / or block diagram.

[0084] Computer-readable program instructions can 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, thereby causing the instructions that execute on the computer, other programmable data processing apparatus, or other device to perform the functions / actions specified in one or more boxes of a flowchart and / or block diagram.

[0085] The flowcharts and block diagrams in the accompanying figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products under various scenarios. In this respect, each block in a flowchart or block diagram may represent a module, segment, or portion of an instruction, which contains one or more executable instructions for implementing the specified logical function. In some alternative implementations, the functions marked in the blocks may occur in a different order than those shown in the figures. For example, two consecutive blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should also be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, can be implemented using a dedicated hardware-based system that performs the specified function or action, or using a combination of dedicated hardware and computer instructions.

[0086] Various examples have been described above. The foregoing descriptions are exemplary and not exhaustive, nor are they limited to the disclosed implementations. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described implementations. The terminology used herein is chosen to best explain the principles, practical applications, or improvements to technology in the market, or to enable others skilled in the art to understand the various implementations disclosed herein.

Claims

1. A method for determining the topological type of a quantum system, comprising: By using a neural network for quantum systems, multiple quantum states of the quantum system are determined by inserting magnetic flux into the quantum system. The neural network represents the wave function of the quantum system, and the multiple quantum states correspond to the multiple inserted magnetic fluxes. Based on the neural network, the corresponding polarization states of the quantum system under the multiple quantum states are determined; as well as The topology of the quantum system is determined based on the corresponding polarization states under the multiple quantum states.

2. The method of claim 1, wherein determining the plurality of quantum states of the quantum system comprises: Based on the quantum system without magnetic flux insertion, the neural network is trained to obtain an initial quantum state as one of the plurality of quantum states, the initial quantum state corresponding to a magnetic flux of zero; as well as The neural network is incrementally updated from the initial quantum state by increasing the magnetic flux inserted into the quantum system to obtain the remaining quantum states other than the initial quantum state among the plurality of quantum states.

3. The method of claim 2, wherein incrementally updating the neural network comprises: For one round of increments in a multi-round increment, Based on the magnetic flux inserted into the quantum system in this round of increment, the Hamiltonian for the quantum system is determined; The energy of the quantum system is determined based on the Hamiltonian and the neural network. as well as The neural network is updated by reducing the energy of the quantum system, and the energy reduction is achieved using the Monte Carlo algorithm.

4. The method of claim 2, wherein the polarization phase torsion increases from zero to an integer multiple of a predetermined angle, the polarization phase corresponding to the increasing magnetic flux inserted into the quantum system.

5. The method of claim 2, wherein the difference between training the neural network and incrementally updating the neural network includes at least one of the following: training step size and learning rate.

6. The method of claim 1, wherein determining the corresponding polarization state of the quantum system in the plurality of quantum states includes For each of the plurality of quantum states, Based on the neural network corresponding to this quantum state, the corresponding positions of multiple electrons in this quantum state are determined, and these multiple electrons belong to the quantum system; and Based on the wave vector in this quantum state and the corresponding positions of the plurality of electrons, the polarization state of the quantum system in this quantum state is determined.

7. The method of claim 1, wherein determining the topological type of the quantum system comprises: Based on the corresponding polarization states under the multiple quantum states, the polarization characteristics of the quantum system are determined, and the polarization characteristics indicate the trend of the polarization of the quantum system as a function of the inserted magnetic flux. as well as The topology type of the quantum system is determined based on the polarization characteristics.

8. The method of claim 7, wherein determining the topological type of the quantum system based on the polarization characteristics comprises: Based on the polarization properties, the topological invariants of the quantum system are determined, and the topological invariants characterize the topological type.

9. The method of claim 1, wherein the direction of the inserted magnetic flux is not parallel to the polarization direction of the corresponding polarization state.

10. An apparatus for determining the topological type of a quantum system, comprising: A quantum state determination module is configured to determine multiple quantum states of a quantum system by inserting magnetic flux into the quantum system using a neural network for the quantum system, wherein the neural network represents the wave function of the quantum system and the multiple quantum states correspond to the multiple inserted magnetic fluxes, respectively. A polarization state determination module is configured to determine the corresponding polarization state of the quantum system under the plurality of quantum states based on the neural network; as well as The topology type determination module is configured to determine the topology type of the quantum system based on the corresponding polarization states under the plurality of quantum states.

11. An electronic device, comprising: At least one processor; as well as At least one memory coupled to the at least one processor and storing 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 9 when executed by the at least one processor.

12. A computer-readable storage medium having stored thereon computer-executable instructions that can be executed by a processor to implement the method according to any one of claims 1 to 9.

13. A computer program product tangibly stored in a computer storage medium and comprising computer-executable instructions that, when executed by a device, cause the device to perform the method according to any one of claims 1 to 9.

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