BEC ground state and excitation state determination method and device, equipment and storage medium

Abstract

The invention provides a BEC ground state and excitation state determination method and device, equipment and a storage medium. Relates to the technical field of quantum computing. The method comprises the following steps: describing a rotation BEC in a coordinate system by adopting a multi-dimensional dimensionless Gross-Pitaevskii (G-P) equation with an angular momentum rotation item to obtain a representation equation of the rotation BEC, and processing the representation equation of the rotation BEC by adopting a virtual time method and a projection method to obtain a space-time sequence equation of a characteristic value; based on the space-time sequence equation of the characteristic values, using an OC method in the space direction and using a backward Euler method and / or a time midpoint method in the time direction to carry out complete discretization, obtaining a backward Euler type high-order format and a time midpoint type high-order format of the space-time sequence equation, carrying out iterative solution on the high-order formats, and determining a ground state and an excitation state. The device comprises a plurality of modules and is used for executing the method, the equipment and the storage medium comprise a processor, the method is executed through the processor, more calculation space can be saved, and universality is better.

Description

Technical Field

[0001] The present application relates to the field of quantum computing technology, and in particular, to a method, apparatus, device, and storage medium for determining the ground state and excited state of BEC. Background Art

[0002] Bose-Einstein condensate (BEC) is a completely new state of matter, different from the traditional four states of matter (gas, liquid, solid, plasma). Since 1995, BEC has been continuously realized in the laboratory, promoting the development of quantum and cold atom physics research. By using magnetic potential well and optical trap technologies, scientists can cool and trap atoms to form BEC. The optical trap technology endows bosons with spin degrees of freedom, and the BEC composed of such bosons is called rotating BEC. The research on rotating BEC is of great significance for exploring the properties of superfluids and the development of quantum computers. The formation and behavior of rotating BEC not only reveal the unique characteristics of superfluids under rotation, but also provide a new way to study complex quantum phenomena. Condensates of different particle types result in different control equations, which also continuously pose new problems and challenges to the research of numerical methods and quantum control. The manifestation of the rotating BEC problem is that the control equation contains a rotation term of particle motion, and the analytical method has certain limitations in studying such nonlinear problems. The experimental research on it is costly, and the experimental observation is also limited by objective conditions. Therefore, the development of numerical methods for the nonlinear Gross-Pitaevskii (G-P) equation under the action of spin not only has important significance for cold atom physics research, but also has important application value in quantum computers. A quantum computer is a device that encodes and calculates information using quantum states. Due to characteristics such as quantum superposition and quantum entanglement, its operation efficiency can be exponentially improved, and it has a higher computational simulation effect than traditional computers in solving complex problems. The existence of the spin term brings new difficulties to the numerical solution of the rotating BEC problem, so it is necessary to develop a set of efficient numerical methods to solve the nonlinear G-P equation with a rotation term.

[0003] In the prior art, there is a literature that proposes a time-splitting spectral method and a method of minimizing the energy function through finite element approximation to solve the ground state, as well as a normalized gradient flow method to calculate the ground state. The normalized gradient flow method mainly includes solving the gradient flow equation and projecting the solution at each time step onto the unit sphere, and discusses different discrete formats of the gradient flow equation, using a finite difference format or a spectral method for discretization in space and a forward (backward) Euler format or a Crank-Nicolson format for discretization in time.

[0004] However, the time-splitting spectral method has advantages in dealing with time-evolution problems but is not suitable for dealing with static ground-state problems; while the finite element method is highly versatile but has high computational complexity, is complex to implement, and has high requirements for handling boundary conditions. The difference method is the most commonly used method, but all existing methods for solving the ground state are of second-order accuracy and there is no high-order accuracy format. Moreover, the energy gap between quantum states is too small, resulting in the possible inability to achieve quantum error correction. In addition, the depth of the corresponding quantum circuit for this type of method is very shallow and long-range quantum logic gates cannot be implemented, leading to poor algorithm performance. Summary of the Invention

[0005] The present application provides a method, apparatus, device, and storage medium for determining the ground state and excited state of BEC. By using the backward Euler-type high-order format and the time midpoint-type high-order operator compensation format to solve the ground-state problem of rotating BEC, the present application is also applicable to the calculation of quantum problems without a rotation term, and a fast iterative processing method is designed in the solution, which can save more computational space and has better versatility.

[0006] In a first aspect, the present application provides a method for determining the ground state and excited state of BEC, including:

[0007] Describing the rotating BEC in a coordinate system by using a multi-dimensional dimensionless G-P equation with an angular momentum rotation term to obtain a representation equation of the rotating BEC, where the representation equation of the rotating BEC includes an external potential function, a wave function, a rotation operator, and a rotation frequency along the axial direction;

[0008] Based on the representation equation of the rotating BEC, determining a target problem for solving the ground state; wherein, the target problem is to determine the eigenfunction corresponding to the lowest energy required under the condition that the wave function satisfies the constraint or to determine the wave function corresponding to the minimum energy on the unit sphere;

[0009] Using the imaginary-time method and the projection method to process the representation equation of the rotating BEC to obtain a spatio-temporal sequence equation of eigenvalues;

[0010] Based on the spatio-temporal sequence equation of eigenvalues, performing full discretization using the OC method in the spatial direction and the backward Euler method in the time direction to obtain a backward Euler-type high-order format of the spatio-temporal sequence equation, and / or based on the spatio-temporal sequence equation of eigenvalues, performing discretization using the OC method and the time midpoint method in the spatial direction and the time direction respectively to obtain a time midpoint-type high-order format of the spatio-temporal sequence equation;

[0011] Based on the target problem, performing iterative solution on the backward Euler-type high-order format of the spatio-temporal sequence equation and / or the time midpoint-type high-order format of the spatio-temporal sequence equation to determine the ground state;

[0012] Calculating the excited state according to the determined ground state.

[0013] Second aspect, the present application provides a BEC ground state and excited state determination device, the device comprising:

[0014] A rotating BEC description module, configured to describe the rotating BEC in a coordinate system by using a multi-dimensional dimensionless Gross-Pitaevskii equation with an angular momentum rotation term, and obtain a representation equation of the rotating BEC, the representation equation of the rotating BEC including an external potential function, a wave function, a rotation operator, and a rotation frequency along the axial direction;

[0015] A target problem determination module, configured to determine a target problem for solving the ground state based on the representation equation of the rotating BEC; wherein, the target problem is to determine an eigenfunction corresponding to the lowest energy required when the wave function satisfies the constraint condition or to determine a wave function corresponding to the minimum energy on the unit sphere;

[0016] A preprocessing module, configured to process the representation equation of the rotating BEC by using the imaginary time method and the projection method to obtain a spatio-temporal sequence equation of eigenvalues;

[0017] A high-order format acquisition module, configured to perform complete discretization in the spatial direction by using the OC method and in the temporal direction by using the backward Euler method based on the spatio-temporal sequence equation of eigenvalues to obtain a backward Euler type high-order format of the spatio-temporal sequence equation, and / or to perform discretization in the spatial direction and the temporal direction by using the OC method and the time midpoint method respectively based on the spatio-temporal sequence equation of eigenvalues to obtain a time midpoint type high-order format of the spatio-temporal sequence equation;

[0018] A ground state solving module, configured to perform iterative solution on the backward Euler type high-order format of the spatio-temporal sequence equation and / or the time midpoint type high-order format of the spatio-temporal sequence equation based on the target problem to determine the ground state;

[0019] An excited state calculation module, configured to calculate the excited state according to the determined ground state.

[0020] Third aspect, an embodiment of the present application provides an electronic device, comprising: at least one processor and a memory; the memory stores computer execution instructions; the at least one processor executes the computer execution instructions stored in the memory, so that the at least one processor executes the BEC ground state and excited state determination method as described in the first aspect and various possible designs of the first aspect above.

[0021] Fourth aspect, an embodiment of the present application provides a computer-readable storage medium, in which computer execution instructions are stored, and when a processor executes the computer execution instructions, the BEC ground state and excited state determination method as described in the first aspect and various possible designs of the first aspect above is implemented.

[0022] Fifth aspect, an embodiment of the present application provides a computer program product, including a computer program, which when executed by a processor, implements the method for determining the BEC ground state and excited state as described in the first aspect above and various possible designs of the first aspect.

[0023] The method, device, equipment and storage medium for determining the BEC ground state and excited state provided by the present application have at least the following beneficial effects:

[0024] 1. After spatio-temporal discretization, a non-linear algebraic equation set can be obtained. The numerical difficulty in solving high-dimensional problems is how to effectively calculate large-scale non-linear equation sets. The solution matrix obtained from high-dimensional space discretization is usually very large and sparse. For traditional computers, the CPU cost and storage capacity of such matrices are difficult to handle high-dimensional problems. The present application develops a high-precision and high-efficiency numerical method for solving the BEC ground state based on the imaginary-time method, enabling high-precision calculations in numerical simulations. The research on efficient numerical methods provides new algorithmic technical support for BEC research and has important application value.

[0025] 2. In order to solve the multi-dimensional rotating BEC equation, the present application proposes two high-order numerical formats for solving the rotating BEC ground state, namely the OCBE format and the OCTM format. Both of these formats have arbitrary even-order accuracy in the spatial direction. To simplify the solution process and reduce the computational amount, the present invention develops an iterative solution method for the two high-order numerical formats. This method shows high efficiency in actual calculations, especially when dealing with the rotating BEC ground state under different external potential conditions. Numerical examples verify that both of these high-order numerical formats can effectively solve the Bose-Einstein condensate ground state with zero and non-zero rotating terms under different external potential conditions.

[0026] 3. The high-order numerical formats and iterative solution methods proposed by the present application have broad potential in various application fields. They are not only applicable to the high-precision solution of the quantum states of rotating Bose-Einstein condensates (BECs) under different external potential conditions, but also have important application value in the fields of quantum computers, numerical simulations, computational physics, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] The drawings here are incorporated into the description and form a part of this description, showing embodiments consistent with the present application, and are used together with the description to explain the principles of the present application.

[0028] Figure 1 is a flowchart of a method for determining the BEC ground state and excited state provided by an embodiment of the present application;

[0029] Figure 2 is a schematic diagram of the evolution of energy with time in the OCBE format provided by an embodiment of the present application;

[0030] Figure 3 Schematic diagram of the ground state solution in OCBE format provided by the embodiments of this application;

[0031] Figure 4 Schematic diagram of the evolution of energy over time in OCTM format provided by the embodiments of this application;

[0032] Figure 5 Schematic diagram of the ground state solution in OCTM format provided by the embodiments of this application;

[0033] Figure 6 Schematic diagram of the evolution of energy over time in OCBE format provided by the embodiments of this application;

[0034] Figure 7 Schematic diagram of the first excited state solution in OCBE format provided by the embodiments of this application;

[0035] Figure 8 Schematic diagram of the evolution of energy over time in OCTM format provided by the embodiments of this application;

[0036] Figure 9 Schematic diagram of the first excited state solution in OCTM format provided by the embodiments of this application;

[0037] Figure 10 Schematic diagram of the evolution of energy over time in OCBE format provided by the embodiments of this application;

[0038] Figure 11 Schematic diagram of the ground state solution in OCBE format provided by the embodiments of this application;

[0039] Figure 12 Schematic diagram of the evolution of energy over time in OCTM format provided by the embodiments of this application;

[0040] Figure 13 Schematic diagram of the ground state solution in OCTM format provided by the embodiments of this application;

[0041] Figure 14 Schematic diagram of the evolution of energy over time in OCBE format provided by the embodiments of this application;

[0042] Figure 15 Schematic diagram of the ground state solution in OCBE format provided by the embodiments of this application;

[0043] Figure 16 Schematic diagram of the evolution of energy over time in OCTM format provided by the embodiments of this application;

[0044] Figure 17 Schematic diagram of the ground state solution in OCTM format provided by the embodiments of this application;

[0045] Figure 18 Schematic diagram of the evolution of energy in the OCBE format provided by the embodiments of the present application over time;

[0046] Figure 19 Schematic diagram of the ground state solution in the OCBE format provided by the embodiments of the present application;

[0047] Figure 20 Schematic diagram of the evolution of energy in the OCTM format provided by the embodiments of the present application over time;

[0048] Figure 21 Schematic diagram of the ground state solution in the OCTM format provided by the embodiments of the present application;

[0049] Figure 22 Schematic diagram of the evolution of energy in the OCBE format provided by the embodiments of the present application over time;

[0050] Figure 23 Schematic diagram of the ground state solution in the OCBE format provided by the embodiments of the present application;

[0051] Figure 24 Schematic diagram of the evolution of energy in the OCTM format provided by the embodiments of the present application over time;

[0052] Figure 25 Schematic diagram of the ground state solution in the OCTM format provided by the embodiments of the present application;

[0053] Figure 26 Schematic diagram of the evolution of energy in the OCBE format provided by the embodiments of the present application over time;

[0054] Figure 27 Schematic diagram of the ground state solution in the OCBE format provided by the embodiments of the present application;

[0055] Figure 28 Schematic diagram of the evolution of energy in the OCTM format provided by the embodiments of the present application over time;

[0056] Figure 29 Schematic diagram of the ground state solution in the OCTM format provided by the embodiments of the present application;

[0057] Figure 30 Schematic diagram of the ground state surface in the OCBE format provided by the embodiments of the present application, Υ = 0.2;

[0058] Figure 31 Schematic diagram of the ground state surface in the OCTM format provided by the embodiments of the present application, Υ = 0.2;

[0059] Figure 32 Schematic diagram of the ground state plane cloud map in the OCBE format provided by the embodiments of the present application, Υ = 0.2;

[0060] Figure 33The ground state planar cloud map in OCTM format provided by the embodiment of this application, Υ = 0.2;

[0061] Figure 34 The schematic diagram of the ground state surface in OCBE format provided by the embodiment of this application, Υ = 0.5;

[0062] Figure 35 The schematic diagram of the ground state surface in OCTM format provided by the embodiment of this application, Υ = 0.5;

[0063] Figure 36 The ground state planar cloud map in OCBE format provided by the embodiment of this application, Υ = 0.5;

[0064] Figure 37 The ground state planar cloud map in OCTM format provided by the embodiment of this application, Υ = 0.5;

[0065] Figure 38 The schematic diagram of the ground state surface in OCBE format provided by the embodiment of this application, Υ = 0.7;

[0066] Figure 39 The schematic diagram of the ground state surface in OCTM format provided by the embodiment of this application, Υ = 0.7;

[0067] Figure 40 The ground state planar cloud map in OCBE format provided by the embodiment of this application, Υ = 0.7;

[0068] Figure 41 The ground state planar cloud map in OCTM format provided by the embodiment of this application, Υ = 0.7;

[0069] Figure 42 The schematic diagram of the structure of a BEC ground state and excited state determination device provided by the embodiment of this application.

[0070] Through the above-mentioned drawings, the specific embodiments of this application have been shown, and there will be more detailed descriptions hereinafter. These drawings and textual descriptions are not intended to limit the scope of the concept of this application in any way, but to illustrate the concept of this application to those skilled in the art by referring to specific embodiments. Detailed implementation manners

[0071] Here, the exemplary embodiments will be described in detail, and the examples are shown in the drawings. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The implementation manners described in the following exemplary embodiments do not represent all the implementation manners consistent with this application. On the contrary, they are merely examples of the devices and methods consistent with some aspects of this application as detailed in the appended claims.

[0072] In the technical solution of this application, the processing of data and other information, such as collection, storage, use, processing, transmission, provision, and disclosure, complies with the provisions of relevant laws and regulations and does not violate public order and good customs.

[0073] It should be noted that in the embodiments of this application, some existing solutions in the industry, such as certain software, components, models, etc., may be mentioned. They should be regarded as exemplary. The purpose is only to illustrate the feasibility in the implementation of the technical solution of this application, but it does not mean that the applicant has already or necessarily used this solution.

[0074] The following will specifically describe the technical solution of this application and how the technical solution of this application solves the above technical problems in detail through specific embodiments. These several specific embodiments can be combined with each other, and the same or similar concepts or processes may not be repeated in some embodiments. The embodiments of this application will be described below with reference to the accompanying drawings.

[0075] Embodiment 1:

[0076] The research on rotating Bose-Einstein condensates (BECs) is not only crucial for exploring the properties of superfluids but also shows significant application value in the development of quantum computers. The research in this field focuses on revealing the unique behavior of superfluids under rotational conditions while opening up new paths for understanding and controlling complex quantum phenomena. The mathematical description of rotating BECs includes a rotational term for particle motion, and the use of frequency-domain parameters enables more efficient implementation of quantum control and quantum error correction, and corresponding quantum states are generated through quantum computers. Numerical simulation plays an important role in the research of BECs. Currently, the accuracy of numerical difference schemes for solving the ground state and excited states of BECs is generally not high, and the quantum error correction ability is weak. Therefore, the embodiments of this application propose a method for determining the ground state and excited states of BECs to provide a more efficient and high-precision numerical format, and this method can effectively eliminate numerical noise. The method for determining the ground state and excited states of BECs is a set of efficient numerical methods specifically designed to solve the quantum states of the nonlinear Gross-Pitaevskii equation with a rotational term. This method is energy-decreasing and can also be used to eliminate numerical noise. Overall, the backward Euler-type high-order format (OCBE) and the time midpoint-type high-order operator compensation format (OCTM) are adopted. To simplify the calculation process and reduce the computational resource requirements, an iterative solution method is further developed, significantly improving the computational efficiency, especially when dealing with the ground state and excited state problems of rotating BECs under different external potential conditions.

[0077] Specifically, as Figure 1 shown, it is a flowchart of a method for determining the ground state and excited states of BECs provided by the embodiments of this application. The method for determining the ground state and excited states of BECs includes the following steps S10 - S60.

[0078] S10. Use the multi-dimensional dimensionless G-P equation with an angular momentum rotation term to describe the rotating BEC in the coordinate system, and obtain the representation equation of the rotating BEC. The representation equation of the rotating BEC includes the external potential function, the wave function, the rotation operator, and the rotation frequency along the axial direction.

[0079] S20. Based on the representation equation of the rotating BEC, determine the target problem for solving the ground state; wherein, the target problem is to determine the eigenfunction corresponding to the lowest energy obtained when the wave function satisfies the constraint condition, or to determine the wave function corresponding to the minimum energy on the unit sphere.

[0080] S30. Use the imaginary-time method and the projection method to process the representation equation of the rotating BEC, and obtain the spatio-temporal sequence equation of the eigenvalues.

[0081] S40. Based on the spatio-temporal sequence equation of the eigenvalues, perform full discretization using the OC method in the spatial direction and the backward Euler method in the time direction to obtain the backward Euler type high-order format of the spatio-temporal sequence equation, and / or based on the spatio-temporal sequence equation of the eigenvalues, perform discretization using the OC method and the time midpoint method in the spatial direction and the time direction respectively to obtain the time midpoint type high-order format of the spatio-temporal sequence equation.

[0082] S50. Based on the target problem, perform iterative solution on the backward Euler type high-order format of the spatio-temporal sequence equation and / or the time midpoint type high-order format of the spatio-temporal sequence equation to determine the ground state.

[0083] S60. Calculate the excited state according to the determined ground state.

[0084] The following will fully expand the above six steps, namely steps S10 - S60, and explain in detail the data processing process and its principle of each step.

[0085] In step S10, according to the mean field theory, when the temperature is much lower than the critical temperature, the rotating BEC in the coordinate system can be effectively described by the multi-dimensional dimensionless G-P equation with an angular momentum rotation term, as follows:

[0086]

[0087] where i is the imaginary unit, x is the spatial coordinate, t is the time, β is a real number and is expressed as a dimensionless parameter, Δ represents the Laplace operator, V(x) is the external potential function, and its frequency in each direction is γ d , is the rotation operator, Υ is the rotation frequency along the axial direction, d = 1, 2, 3 represents the spatial dimension. The wave function ψ(x, t) satisfies mass conservation:

[0088]

[0089] and energy conservation:

[0090]

[0091] wherein is the conjugate of ψ(x,t).

[0092] In step S20, the purpose is to determine the target problem, so as to iteratively solve the backward Euler type high-order format and / or the time midpoint type high-order format in subsequent steps according to the determined target problem, thereby obtaining the ground state solution.

[0093] Specifically, this embodiment details the related content about the ground state of the rotating BEC. The ground state of the rotating BEC is the wave function φ g (x) that satisfies the constrained equation for the nonlinear eigenvalue problem:

[0094]

[0095] Then the eigenvalue μ can be expressed by its corresponding eigenfunction φ(x) as:

[0096]

[0097] where E Υ (φ) is the energy function related to φ given in equation (3).

[0098] Under the condition of satisfying the constraint , the eigenfunction φ(x) corresponding to the lowest energy sought is the ground state of the rotating BEC. The ground state is also the wave function corresponding to the minimum energy on the unit sphere S = {φ|||φ||2 = 1, E Υ (φ) < ∞}, that is, to find φ g0 ∈S such that

[0099]

[0100] In step S30, in order to find the ground state solution of the rotating BEC, this embodiment uses a combination of the imaginary time method and the projection method to deal with equation (1), and the steps are as follows:

[0101] Set the time step as Δt > 0, and divide the time direction into t n = nΔt (n = 0, 1, 2,...), then in each time interval [t n , t n+1 there is

[0102]

[0103] wherein The initial conditions of equations (7)-(8) are

[0104] φ(x, 0) = φ0(x), x ∈ R d , (9)

[0105] where the wave function φ(x, t) satisfies the normalization condition

[0106]

[0107] Since the G - P equation with the angular momentum rotation term has homogeneous Dirichlet boundary conditions, the spatial domain of Eqs. (7)-(8) can be truncated from R d to the bounded region Ω = [a, b] × [c, d].

[0108]

[0109] The boundary conditions and initial conditions can be set as

[0110]

[0111] φ(x, 0) = φ0(x), x ∈ Ω and ||φ0||2 := ∫ Ω |φ0(x)| 2 dx = 1, (13)

[0112] where φ0(x) is a known function.

[0113] The vortex center in multi - dimensional rotating BEC is just a point and the geometric structure is relatively simple.

[0114] In steps S40 and S50, this embodiment takes two - dimensional calculation as an example to introduce in detail the method for calculating the ground state of multi - dimensional rotating BEC.

[0115] Select positive integers J and K to represent the total number of partitions in the x and y directions of space respectively. The spatial grid step size is h x = (b - a) / J, h y = (d - c) / K, and the spatial grid nodes are (x j , y k ) = (a + jh x , c + kh y ), j = 0, 1,..., J, k = 0, 1,..., K. Set the set of node subscripts as:

[0116] Ω H = {(j, k)|j = 1, 2,..., J - 1, k = 1, 2,..., K - 1},

[0117]

[0118] where Select the time step size τ and represent the time nodes as tn := nτ, (n = 1, 2,...), using to represent the numerical approximation of φ(x, t) at (x j , y k , t n ). There is a first-order partial derivative in Equation (10). Similarly, the high-order operator compensation method can be used to approximately construct a high-order approximation of the first-order partial derivative. First, introduce the following difference operators:

[0119]

[0120] Through the Taylor expansion formula, the difference operator can be expressed as:

[0121]

[0122] In Equation (14), using the high-order operator compensation method, a high-order approximation operator of can be obtained. Let

[0123] c k (k = 1, 2,..., ω) be constants, and define the difference operator as

[0124]

[0125] Then, the high-order approximation of is

[0126]

[0127] For any high-order approximation of Equation (16), through the Taylor expansion formula and some algebraic operations, the expression of the coefficient c k (k = 1, 2, 3,....) can be obtained:

[0128]

[0129] For the sake of easy understanding, when constructing 2nd-order, 4th-order, and 6th-order accuracy methods, the specific values of c k are shown in Table 1.

[0130] Table 1 Coefficients of the OC method

[0131] Order <![CDATA[c1]]> <![CDATA[c2]]> … 2-nd, ω=0 - - - 4-th, ω=1 -1 / 6 - - 6-th, ω=2 -1 / 6 1 / 30 - … … … …

[0132] where can be specifically expressed as

[0133] Similarly, when ω = 2, using the high-order operator compensation method, a high-order approximation operator of is

[0134]

[0135] Among them

[0136]

[0137] The coefficient c of Equation (19) k (k = 1, 2, 3,....) has the same expression as Equation (17), and the specific values are shown in Table 1 above.

[0138] Then, the OC method is used for the spatial direction of Equation (10) and the backward Euler method is used for its temporal direction for full discretization, and the fully discrete high-order format for solving Equation (10) - the backward Euler type high-order format, also known as the OCBE format, is as follows:

[0139]

[0140] Among them

[0141]

[0142] Here

[0143]

[0144] The discrete initial conditions and boundary conditions are respectively:

[0145]

[0146] At the end of each time step, for a normalization process is performed, that is, let

[0147]

[0148] Among them, the two-dimensional discrete norm The discrete energy and eigenvalues of Equation (20) can be expressed as:

[0149]

[0150] Among them is the conjugate of. Similarly, Equation (20) can be numerically calculated using the linearization method and the iterative method respectively. In actual calculations, the OCBE format using the linearization method can be written as:

[0151]

[0152] At this time, although it is converted to solving a simpler system of linear equations, in the actual process of solving Equation (25), the order of the coefficient matrix of this system of equations is (J - 1)*(K - 1). When the grid is relatively coarse, that is, when J and K take small values, the Gauss - Seidel iteration method can still be used for calculation. However, if the grid points become finer and the solution matrix is too large, the computer memory will crash and the solution cannot be carried out. Therefore, in this embodiment, a more practical iterative solution method is designed to handle this problem, and the specific steps are as follows:

[0153]

[0154] In the formula

[0155]

[0156] When m = 0, that is, at the initial value of the iteration, it can be set as

[0157]

[0158] At this time, regardless of the value of ω, the coefficient matrix for solving the above formula (26) can always maintain a standard block upper triangular matrix, and thus can be solved quickly. The termination condition of the iteration is restricted. Given ε as the error threshold, that is, for a certain positive integer m, there is

[0159] Then take

[0160] Similarly, using the OC method and the time - midpoint method to discretize the spatial direction and time direction of Equation (10) respectively, another fully - discretized high - order format for solving Equation (10) can be obtained - the time - midpoint type high - order format, namely the OCTM format, and its form is as follows:

[0161]

[0162] Among them V j,k :=V(x j ,y k ), the operator is the same as that in Equation (20). The initial conditions and boundary conditions are the same as those in (21)-(22). Similarly, after each time step ends, it is necessary to perform unit normalization processing, that is, let

[0163]

[0164] Here, the discrete energy of the numerical format OCTM can be calculated using Equation (24).

[0165] Equation (31) can also be numerically calculated using the linearization processing method and the iterative processing method respectively. In actual calculation, the OCTM format after using the linearization processing method can be written as:

[0166]

[0167] All other conditions and projection steps are the same as the original OCTM format (30).

[0168] Obviously, Equation (32) is a simple system of linear equations. Only when the grid points are relatively coarse, the Gauss-Seidel iteration method can be used for calculation. When the grid points become finer, there is also the problem of an overly large solution matrix, and the memory of the computer will limit the use of the Gauss-Seidel iteration method for solution. Therefore, in this embodiment, an iterative solution method is designed for calculation, and the specific steps are as follows:

[0169]

[0170] In the formula There is And Similar to Equations (27) and (28), the initial value of the iteration is the same as Equation (29). It can be seen that regardless of the value of ω, the coefficient matrix of Equation (33) is always a standard block upper triangular matrix, and it can be solved quickly. Similarly, in actual solution, a given error threshold is set to limit the termination condition of the iteration.

[0171] In step S60, for the first excited state φ g1 , its energy is ranked second lowest. As long as the initial iteration value is orthogonal to the ground state, that is, an orthogonal vector is selected as the initial value in the Hilbert space, and then the above ground state solution method is used for iteration, the first excited state and the corresponding energy and eigenvalue μ g1 can be obtained. By analogy, the quantum states φ gl of the required excited states and the corresponding energies E gl (l = 2, 3,...) and eigenvalues μ gl (l = 2, 3,...) can be obtained.

[0172] After step S60, the method further includes:

[0173] Define the quantum state energy gap as:

[0174] ΔE gl = E gl - E g(l-1) (l = 2, 3,...)

[0175] Where ΔE gl is the quantum state energy gap,

[0176] and prepare the quantum state through a parameterized quantum circuit and use the parameters (frequency domain and rotation intensity) to optimize and adjust the energy gap to be as large as possible. So that quantum error correction is easy to implement.

[0177] As long as the external environment remains unchanged, any quantum wave function ψ(x) can be expressed as a superposition of the ground state β l and the excited state quantum state φ gl i.e.,

[0178]

[0179] where M is a positive integer that can be set according to the precision of quantum computing and control.

[0180] Example 2:

[0181] Based on the BEC ground state and excited state determination method provided in Example 1, in the case of rotating Bose-Einstein condensate, two high-order numerical formats for solving the ground state and excited state of rotating Bose-Einstein condensate are used, namely the backward Euler-type high-order format and the time midpoint-type high-order format. Both have arbitrary even-order precision in the spatial direction, and as time evolves, the energy gradually decays. In the actual solution process, an iterative solution method is developed for the two high-order numerical formats to simplify the solution process and reduce the computational amount. To prove that the present application can achieve the above effects, this example provides a numerical example based on the method proposed in this article. Through the numerical example, it is verified that under different external potentials, both high-order numerical formats can effectively solve the ground state and excited state of Bose-Einstein condensate with zero and non-zero rotation terms.

[0182] Next, in this example, the external potential function is taken as the one-dimensional harmonic oscillator potential V(x) = x 2 / 2 to test the spatial precision of the two numerical formats when solving the ground state.

[0183] The initial conditions are selected as The calculation interval range is Ω = [-16, 16] and β = 60. Since there is no exact analytical solution, the fine grid numerical solution is taken as the exact solution. Here, when h = 1 / 128 and τ = 10 -4 then there is E g0 = E(φ g0 ) = 6.0759452. Let the ground state energy error be e = |E g - E g0 |, then e BE and e TM respectively represent the energy errors of the ground state obtained by the OCBE format and the OCTM format at different step sizes; the L ∞ norm error of the solution is denoted as ||e|| ∞ = max|φ g - φ g0 |, let and respectively represent the maximum errors between the ground state wave functions obtained by the two formats at different step sizes and the exact solution. t BEIndicates the solution cut-off time. See Tables 2 to 5.

[0184] Table 2 Spatial error analysis of the ground state solved by the OCBE format when the external potential function is the one-dimensional harmonic oscillator potential and ω = 1

[0185]

[0186]

[0187] Table 3 Spatial error analysis of the ground state solved by the OCTM format when the external potential function is the one-dimensional harmonic oscillator potential and ω = 1

[0188]

[0189] Table 4 Spatial error analysis of the ground state solved by the OCBE format when the external potential function is the one-dimensional harmonic oscillator potential and ω = 2

[0190]

[0191] Table 5 Spatial error analysis of the ground state solved by the OCTM format when the external potential function is the one-dimensional harmonic oscillator potential and ω = 2

[0192]

[0193] Next, the harmonic oscillator potential + optical particle potential V(x) = x 2 / 2 + 25sin 2 (πx / 4) will be selected as the external potential function for numerical calculation to further verify the effectiveness and generality of the proposed high-order numerical formats OCBE and OCTM for solving the ground state and the first excited state problems; and to display the images of the ground state and the first excited state. The spatial precision of both numerical formats is 6th order, and the iterative solution method is used for both calculations. The initial conditions are selected as to calculate the ground state, to calculate the first excited state, and the calculation interval range is Ω = [-16, 16]. See Figures 2 to 9 .

[0194] After that, this embodiment gives the corresponding numerical results, energy evolution, and ground state diagrams of the BEC ground state with zero rotation term calculated using the two high-order numerical formats under different external potentials. To unify the variables, the discrete energy threshold and the iterative error threshold are both set to 10 -6 . The two high-order numerical formats use the format when ω = 2, and the iterative solution method is used for both calculations. To quantify the numerical solution, the widths of the two-dimensional BEC in the x and y directions can be calculated by the following formulas respectively:

[0195]

[0196] First, take the external potential function as the two-dimensional harmonic oscillator potential Calculate the ground state. Select the initial conditions as Let Υ = 0 and β = 80, and calculate the ground state at two different frequencies.

[0197] Here, take the external potential frequency as γ x = γ y = 1. Select the calculation interval range Ω = [-8, 8] 2 , with a step size of h x = h y = 1 / 4 and τ = 10 -3 . The ground state energy E(φ g ) calculated using the OCBE format is 3.5639378, and the total time required to reach the ground state is t BE = 1.511. The widths of the ground state wave function on the x-axis and y-axis are both 1.3271599; the ground state energy E(φ g ) calculated using the OCTM format is 3.5638841, and the total time required to reach the ground state is t TM = 1.471. The widths of the ground state wave function on the x-axis and y-axis are both 1.3282823. The energy evolution and ground state results under the two different numerical formats are as Figures 10 to 13 shown.

[0198] Next, take γ x = 1, γ y = 4. Select the calculation interval range Ω = [-8, 8]×[-4, 4], h x = 1 / 4, h y = 1 / 8 and τ = 10 -3 . The ground state energy E(φ g ) calculated using the OCBE format is 7.4285016, and the total time required to reach the ground state is t BE = 1.604. The widths of the ground state wave function on the x-axis and y-axis are 1.7856870 and 0.8928435 respectively; the ground state energy E(φ g ) calculated using the OCTM format is 7.4282432, and the total time required to reach the ground state is t TM = 1.537. The widths of the ground state wave function on the x-axis and y-axis are 1.7895564 and 0.8947782 respectively. The energy evolution and ground state results under the two different numerical formats are as Figures 14 to 17 shown.

[0199] This embodiment gives the numerical results, energy evolution, and ground state diagrams corresponding to the ground state of BEC with zero rotation term calculated using two high-order numerical formats under different external potentials.

[0200] First, take the external potential function as the two-dimensional harmonic oscillator potential Calculate the ground state. Select the initial conditions as Let Υ = 0 and β = 80, and calculate the ground state at two different frequencies.

[0201] When taking γ x = 1, γ y = 1. Select the calculation interval range Ω = [-8, 8] 2 , with a step size of h x = h y = 1 / 4 and τ = 10 -3 . The ground state energy E(φ g ) calculated using the OCBE format is 3.5639378, and the total time required to reach the ground state is t BE = 1.511. The widths of the ground state wave function on the x-axis and y-axis are both 1.3271599; the ground state energy E(φ g ) calculated using the OCTM format is 3.5638841, and the total time required to reach the ground state is t TM = 1.471. The widths of the ground state wave function on the x-axis and y-axis are both 1.3282823. The energy evolution and ground state results under the two different numerical formats are as Figures 18 to 21 shown.

[0202] When taking γ x = 1, γ y = 4. Select the calculation interval range Ω = [-8, 8]×[-4, 4], h x = 1 / 4, h y = 1 / 8 and τ = 10 -3 . The ground state energy E(φ g ) calculated using the OCBE format is 7.4285016, and the total time required to reach the ground state is t BE = 1.604. The widths of the ground state wave function on the x-axis and y-axis are 1.7856870 and 0.8928435 respectively; the ground state energy E(φ g ) calculated using the OCTM format is 7.4282432, and the total time required to reach the ground state is t TM = 1.537. The widths of the ground state wave function on the x-axis and y-axis are 1.7895564 and 0.8947782 respectively. The energy evolution and ground state results under the two different numerical formats are as Figures 22 to 25 shown.

[0203] From the numerical data, it can be clearly seen that the magnitude of the frequency affects the widths of the ground state in different directions.

[0204] Then take the external potential function as the two-dimensional harmonic oscillator potential and the far blue-detuned Gaussian laser beam corresponding to the stirring potential To calculate the ground state, the external potential can generate vortices in the BEC.

[0205] Select the initial conditions as The calculation interval range is Ω = [-8, 8] 2 . Let Υ = 0, β = 100, γ x = 1, γ y = 1, δ = r0 = 1, the grid size and time step are h x = h y = 1 / 4 and τ = 10 -3 . The ground state energy E(φ g ) calculated using the OCBE format is 4.4346166, and the total time required to reach the ground state is t BE = 1.571. The widths of the ground state wave function on the x-axis and y-axis are both 1.4526915; the ground state energy E(φ g ) calculated using the OCTM format is 4.4345508, and the total time required to reach the ground state is t TM = 1.524. The widths of the ground state wave function on the x-axis and y-axis are both 1.4541135; the energy evolution and ground state results under two different numerical formats are as Figures 26 to 29 shown.

[0206] It can be seen from the above two examples that under different external potentials, the two high-order numerical formats have good application results for calculating the ground state of BEC with a zero rotation term.

[0207] Finally, in this embodiment, the ground state solutions under different rotation strength coefficients Υ are calculated using two high-order numerical formats, and the surface plots and cloud plots of the corresponding ground states are given.

[0208] First, take the external potential function as the two-dimensional harmonic oscillator potential V(x, y) = (x 2 + y 2 ) / 2 to calculate the ground state of the BEC with a non-zero rotation term. Select the initial conditions as:

[0209]

[0210] where The calculation interval range is Ω = [-6, 6] 2 . Set β = 100, h x = h y = 1 / 4 and τ = 10 -3 .

[0211] When Υ = 0.2 is taken, the total time required for calculation using the OCBE format is t BE= 13.635, the widths of the ground-state wave function on the x-axis and y-axis are both 1.4045074; the total time required to calculate the ground state using the OCTM format is t TM = 13.376, the widths of the ground-state wave function on the x-axis and y-axis are both 1.4044600; the ground-state surface plots calculated using two high-order numerical formats are as shown in Figure 30 and Figure 31 shown, and the ground-state cloud plots are as shown in Figure 32 and Figure 33 shown.

[0212] When Υ = 0.5, the total time required to calculate using the OCBE format is t BE = 68.338, the widths of the ground-state wave function on the x-axis and y-axis are both 1.4772157; the total time required to calculate the ground state using the OCTM format is t TM = 68.207, the widths of the ground-state wave function on the x-axis and y-axis are both 1.4771459; the ground-state surface plots calculated using two high-order numerical formats are as shown in Figure 34 and Figure 35 shown, and the ground-state cloud plots are as shown in Figure 36 and Figure 37 shown.

[0213] Select Υ = 0.7, the total time required to calculate using the OCBE format is t BE = 151.612, the widths of the ground-state wave function on the x-axis and y-axis are both 1.6517643; the total time required to calculate the ground state using the OCTM format is t TM = 151.301, the widths of the ground-state wave function on the x-axis and y-axis are both 1.6518926; the ground-state surface plots calculated using two high-order numerical formats are as shown in Figure 38 and Figure 39 shown, and the ground-state cloud plots are as shown in Figure 40 and Figure 41 shown.

[0214] Example 3:

[0215] This embodiment of the present application also provides a BEC ground state and excited state determination device, as shown in Figure 42 shown, the BEC ground state and excited state determination device includes:

[0216] A rotating BEC description module 421, configured to describe the rotating BEC in the coordinate system using a multi-dimensional dimensionless Gross-Pitaevskii equation with an angular momentum rotation term, to obtain a representation equation of the rotating BEC, and the representation equation of the rotating BEC includes an external potential function, a wave function, a rotation operator, and a rotation frequency along the axis;

[0217] A target problem determination module 422, configured to determine a target problem for solving the ground state based on the representation equation of the rotating BEC; wherein, the target problem is to determine the eigenfunction corresponding to the lowest energy obtained when the wave function satisfies the constraint condition or to determine the wave function corresponding to the minimum energy on the unit sphere.

[0218] A preprocessing module 423, configured to process the representation equation of the rotating BEC by using the imaginary time method and the projection method to obtain a spatio-temporal sequence equation of eigenvalues.

[0219] A high-order format acquisition module 424, configured to perform complete discretization in the spatial direction using the OC method and in the temporal direction using the backward Euler method based on the spatio-temporal sequence equation of eigenvalues to obtain a backward Euler type high-order format of the spatio-temporal sequence equation, and / or to perform discretization in the spatial direction and the temporal direction using the OC method and the time midpoint method respectively based on the spatio-temporal sequence equation of eigenvalues to obtain a time midpoint type high-order format of the spatio-temporal sequence equation.

[0220] A ground state solving module 425, configured to perform iterative solution on the backward Euler type high-order format of the spatio-temporal sequence equation and / or the time midpoint type high-order format of the spatio-temporal sequence equation based on the target problem to determine the ground state.

[0221] An excited state calculation module 426, configured to calculate the excited state according to the determined ground state.

[0222] An embodiment of the present application provides an electronic device. The electronic device may include: a processor and a memory, wherein the processor and the memory can communicate; exemplarily, the processor and the memory communicate through a communication bus.

[0223] The processor executes the computer execution instructions stored in the memory, so that the processor executes the solution in the above embodiment. The processor may be a general-purpose processor, including a central processing unit (CPU), a network processor (NP), etc.; it may also be a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components.

[0224] The communication bus can be a Peripheral Component Interconnect (PCI) bus, an Extended Industry Standard Architecture (EISA) bus, or the like. The system bus can be divided into an address bus, a data bus, a control bus, etc. The transceiver is used to implement communication between the database access device and other computers (such as clients, read-write libraries, and read-only libraries). The memory may include Random Access Memory (RAM), and may also include non-volatile memory.

[0225] The electronic device provided by the embodiments of the present application can be the terminal device in the above embodiments.

[0226] The embodiments of the present application also provide a computer-readable storage medium, in which computer instructions are stored. When the computer instructions run on a computer, the computer is caused to execute the technical solutions of the BEC ground state and excited state determination method in the above embodiments.

[0227] The embodiments of the present application also provide a computer program product. The computer program product includes a computer program, which is stored in a computer-readable storage medium. At least one processor can read the computer program from the computer-readable storage medium, and when at least one processor executes the computer program, the technical solutions of the BEC ground state and excited state determination method in the above embodiments can be implemented.

[0228] In several embodiments provided by the present application, it should be understood that the disclosed devices and methods can be implemented in other ways. For example, the device embodiments described above are merely illustrative. For example, the division of modules is only a logical function division. In actual implementation, there may be other division methods. For example, multiple modules can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the displayed or discussed coupling or direct coupling or communication connection between each other can be through some interfaces, and the indirect coupling or communication connection of devices or modules can be in electrical, mechanical or other forms.

[0229] The modules described as separate components may or may not be physically separated, and the components displayed as modules may or may not be physical units, that is, they can be located in one place, or can be distributed to multiple network units. Some or all of the modules can be selected according to actual needs to implement the solution of this embodiment.

[0230] In addition, in each embodiment of the present application, each functional module can be integrated in a processing unit, or each module can exist physically alone, or two or more modules can be integrated in one unit. The unit formed by the above modules can be implemented in the form of hardware, or in the form of a hardware plus a software functional unit.

[0231] The integrated module implemented in the form of a software functional module can be stored in a computer-readable storage medium. The above software functional module stored in a storage medium includes several instructions for causing a computer device (which can be a personal computer, a server, or a network device, etc.) or a processor to execute some steps of the methods in the various embodiments of the present application.

[0232] It should be understood that the above processor can be a Central Processing Unit (CPU for short), or other general-purpose processors, Digital Signal Processors (DSP for short), Application Specific Integrated Circuits (ASIC for short), etc. The general-purpose processor can be a microprocessor or the processor can also be any conventional processor, etc. The steps of the method disclosed in combination with the invention can be directly embodied as being executed and completed by a hardware processor, or executed and completed by a combination of hardware and software modules in the processor.

[0233] The memory may include a high-speed RAM memory, and may also include non-volatile storage NVM, such as at least one disk memory, and can also be a USB flash drive, a mobile hard disk, a read-only memory, a magnetic disk, or an optical disc, etc.

[0234] The bus can be an Industry Standard Architecture (ISA for short) bus, a Peripheral Component Interconnect (PCI for short) bus, or an Extended Industry Standard Architecture (EISA for short) bus, etc. The bus can be divided into an address bus, a data bus, a control bus, etc.

[0235] The above storage medium can be implemented by any type of volatile or non-volatile storage device or a combination thereof, such as static random access memory (SRAM), electrically erasable programmable read-only memory (EEPROM), erasable programmable read-only memory (EPROM), programmable read-only memory (PROM), read-only memory (ROM), magnetic memory, flash memory, magnetic disk or optical disk. The storage medium can be any available medium accessible by a general-purpose or special-purpose computer.

[0236] An exemplary storage medium is coupled to the processor, enabling the processor to read information from the storage medium and write information to the storage medium. Of course, the storage medium can also be a component of the processor. The processor and the storage medium can be located in an application specific integrated circuit (ASIC). Of course, the processor and the storage medium can also exist as discrete components in an electronic control unit or a master control device.

[0237] Those of ordinary skill in the art can understand that all or part of the steps of implementing the above method embodiments can be completed by hardware related to program instructions. The foregoing program can be stored in a computer-readable storage medium. When the program is executed, it performs the steps including the above method embodiments; and the foregoing storage medium includes various media that can store program codes, such as ROM, RAM, magnetic disk or optical disk.

[0238] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application, and not to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A method for determining the ground state and excited state of BEC, characterized in that, The method includes: Describing the rotating BEC in the coordinate system using a multi-dimensional dimensionless G-P equation with an angular momentum rotation term to obtain a representation equation of the rotating BEC, where the representation equation of the rotating BEC includes an external potential function, a wave function, a rotation operator, and a rotation frequency along the axial direction; Based on the representation equation of the rotating BEC, determining a target problem for solving the ground state; wherein, the target problem is to determine the eigenfunction corresponding to the lowest energy required when the wave function satisfies the constraint condition or to determine the wave function corresponding to the minimum energy on the unit sphere; Using the imaginary time method and the projection method to process the representation equation of the rotating BEC to obtain a spatio-temporal sequence equation of eigenvalues; Based on the spatio-temporal sequence equation of eigenvalues, performing full discretization using the OC method in the spatial direction and the backward Euler method in the temporal direction to obtain a backward Euler type high-order format of the spatio-temporal sequence equation, and / or based on the spatio-temporal sequence equation of eigenvalues, performing discretization using the OC method and the time midpoint method in the spatial and temporal directions respectively to obtain a time midpoint type high-order format of the spatio-temporal sequence equation; Based on the target problem, performing iterative solution on the backward Euler type high-order format of the spatio-temporal sequence equation and / or the time midpoint type high-order format of the spatio-temporal sequence equation to determine the ground state; Calculating the excited state according to the determined ground state.

2. The method for determining the BEC ground state and excited state according to claim 1, wherein The representation equation of the rotating BEC is expressed as: where \(i\) is the imaginary unit, \(x=(x,y,z)\) are three-dimensional space coordinates, \(t\) is time, \(\beta\) is a real number and is expressed as a dimensionless parameter, \(\Delta\) represents the Laplace operator, \(V(x)\) is the external potential function, \(\gamma\) d is the frequency of the external potential function in each direction, is the rotation operator, is the partial derivative of the function with respect to time \(t\), \(\Upsilon\) is the rotation intensity coefficient along the \(Z\)-axis, \(d = 1,2,3\) represents the spatial dimension, \(R\) represents the real number space, and \(\psi(x,t)\) is the wave function; The wave function ψ(x,t) satisfies mass conservation: where ψ(x,0) is a continuous function; The wave function ψ(x,t) satisfies energy conservation. The definition and conservation equation of energy are as follows: In the formula is the conjugate of ψ(x,t).

3. The method for determining the BEC ground state and excited state according to claim 2, characterized in that, Based on the representation equation of the rotating BEC, determining a target problem for solving the ground state, including: Determine the ground state of the rotating BEC as the wave function φ g (x) satisfies the nonlinear eigenvalue problem with the constraint equation expressed as: where μ is an eigenvalue; Expressing the eigenvalue μ in terms of the eigenfunction φ(x) corresponding to the eigenvalue μ as: where E Υ (φ) is an energy function related to φ. Subject to the constraint the eigenfunction φ(x) corresponding to the lowest energy sought is the ground state of the rotating BEC, or the ground state is the wave function corresponding to the minimum energy on the unit sphere S = {φ |||φ||2 = 1, E Υ (φ) < ∞}, that is, to find the wave function φ g ∈ S such that where is the minimum energy value, φ g is the ground state, and min is the minimum value function.

4. The method for determining the BEC ground state and excited state according to claim 3, wherein Using the imaginary time method and the projection method to process the representation equation of the rotating BEC to obtain a spatio-temporal sequence equation of eigenvalues, including: Based on the representation equation of the rotating BEC, set the time step as Δt > 0, and divide the time direction into t n = nΔt (n = 0, 1, 2,...), then at each time interval [t n , t n+1 , there is: wherein is The initial conditions of equations (7)-(8) are: φ(x, 0) = φ0(x), x ∈ R d , (9) where φ0(x) is a known function; The wave function φ(x,t) satisfies the normalization condition Truncate the spatial domain of equations (7)-(8) from R d to a bounded region. For two-dimensional problems, take the boundary Ω = [a, b] × [c, d], where a, b, c, and d represent boundary constants respectively; and obtain the spatio-temporal sequence equation of eigenvalues: The boundary conditions and initial conditions of the spatio-temporal sequence equation of eigenvalues are respectively set as: φ(x, 0) = φ0(x), x ∈ Ω and ||φ0||2 := ∫ Ω |φ0(x)| 2 dx = 1. (13) 5. The method for determining the BEC ground state and excited state according to claim 4, characterized in that, Based on the spatio-temporal sequence equation of eigenvalues, performing full discretization using the OC method in the spatial direction and the backward Euler method in the temporal direction to obtain a backward Euler type high-order format of the spatio-temporal sequence equation, and / or based on the spatio-temporal sequence equation of eigenvalues, performing discretization using the OC method and the time midpoint method in the spatial and temporal directions respectively to obtain a time midpoint type high-order format of the spatio-temporal sequence equation, including: Select positive integers \(J\) and \(K\) to represent the total number of sub - divisions in the \(x\) and \(y\) directions of space respectively. The space grid step size is \(h\). x \(=(b - a) / J\), \(h\). y \(=(d - c) / K\), \(h\). x is the step size in the \(x\) - direction, \(h\). y is the step size in the \(y\) - direction. The space grid nodes are \((x\). j , \(y\). k )=(a + jh x , \(c + kh\). y ), \(j = 0,1,\cdots,J\), \(k = 0,1,\cdots,K\). \(x\). j and \(y\). k are the coordinates in the \(x\) and \(y\) directions at the nodes respectively. Set the set \(\Omega\) of interior points with node sub - scripts H and the set of boundary points as follows: Ω H ={(j, k) | j = 1, 2,..., J - 1, k = 1, 2,..., K - 1}, where ω is a positive integer, represents the union of the set of node subscripts and the set of boundary node subscripts, that is Select the time step τ and represent the time nodes as t n := nτ, (n = 1, 2,...), and use to represent the numerical approximation of φ(x, t) at (x j , y k , t n ). Introducing the following difference operators: where represents the x-direction difference operator, is the value of the discrete function at the node ( j±1,k ); By the Taylor expansion formula, the difference operator is expressed as: In the formula is infinitesimals of the same order, which represents the precision. φ(x j , y k , t) is the value of the function at the node (j, k) at time t; In formula (14), a high-order approximation operator of is obtained by using the high-order operator compensation method. Let the coefficient c k (k = 1, 2,..., ω) be a constant, and define the difference operator as: k ​​​ Then we get The high-order approximation of: In the formula, is the value at time t at node (j, k); For any high-order approximation of Equation (16), the expression of coefficient c k (k = 1, 2, 3,....) is obtained through the Taylor expansion formula and algebraic operations: k (k = 1, 2, 3,....) will be specifically represented as where is the difference operator, and φ j±1,k (t) is the value of the function at the nodes (j±1,k); Obtained by using the high-order operator compensation method The high-order approximation operator of is where In the formula, is the y-direction difference operator, is the first-order central difference operator in the y direction; Performing full discretization on the spatial direction of equation (10) using the OC method and its temporal direction using the backward Euler method, the form of the backward Euler type high-order format for solving equation (10) is as follows: where In the formula, is the difference operator in the x direction, is the difference operator in the y direction, V j,k is the value of V(x, y) at the node, a ω is a constant, is the second-order central difference in the y direction, is the forward difference in the y direction, is the backward difference in the y direction; The discrete initial conditions and boundary conditions are respectively: At the end of each time step, for perform unit normalization, that is, let Among them, the two-dimensional discrete norm The discrete energy and eigenvalues of Equation (20) are expressed as: where is the conjugate of, is the discrete energy of the backward Euler type high-order scheme, is the eigenvalue of the backward Euler type high-order scheme; Performing numerical calculations on equation (20) using the linearization processing method and the iterative processing method respectively, the linearization processing method is expressed as: Using the OC method and the time midpoint method to discretize the spatial direction and the temporal direction of equation (10) respectively to obtain another fully discrete high-order format of equation (10) - the time midpoint type high-order format, the form is as follows: wherein V j,k := V(x j , y k ); After each time step, perform normalization on That is, let Calculate the discrete energy of the time midpoint operator compensation numerical format using Equation (24). For Equation (31), numerical calculations are performed using the linearization method and the iterative method respectively. The time midpoint type high-order format after using the linearization method is expressed as:

6. The method for determining the BEC ground state and excited state according to claim 5, characterized in that, Based on the target problem, perform iterative solution on the backward Euler type high-order format of the spatio-temporal sequence equation and / or the time midpoint type high-order format of the spatio-temporal sequence equation to determine the ground state, including: For the backward Euler type high-order format (abbreviated as OCBE), the specific solution process is as follows: In the formula, the operator and are defined as follows: When the initial value of the iteration m = 0, there is: Restrict the termination condition of the iteration. Given an error threshold ε, for a certain positive integer m, there is Then take For the time midpoint type high-order format (abbreviated as OCTM), the specific solution process is as follows: where and represent the (m + 1)-th iteration value, is the n-step value.

7. The method for determining the BEC ground state and excited state according to claim 6, characterized in that According to the determined ground state, calculate the excited state, including: select an orthogonal vector in the Hilbert space as the initial value, and perform iteration using the ground state solution method to obtain the first excited state and the corresponding energy and eigenvalue. By analogy, find the quantum states and the corresponding energies and eigenvalues of each required excited state; The method further includes: determining the quantum state energy gap according to the calculated ground state and excited state, preparing the quantum state through a parameterized quantum circuit, and using the frequency domain and rotation intensity to optimize and adjust the quantum state energy gap to be as large as possible.

8. A BEC ground state and excited state determination device, characterized in that, The device includes: A rotating BEC description module configured to describe the rotating BEC in the coordinate system using the G-P equation with a multi-dimensional dimensionless angular momentum rotation term to obtain the representation equation of the rotating BEC. The representation equation of the rotating BEC includes an external potential function, a wave function, a rotation operator, and a rotation frequency along the axis; A target problem determination module configured to determine the target problem for solving the ground state based on the representation equation of the rotating BEC; wherein, the target problem is to determine the eigenfunction corresponding to the lowest energy required when the wave function satisfies the constraint condition or to determine the wave function corresponding to the minimum energy on the unit sphere; A preprocessing module configured to use the imaginary time method and the projection method to process the representation equation of the rotating BEC to obtain the spatio-temporal sequence equation of the eigenvalue; A high-order format acquisition module configured to, based on the spatio-temporal sequence equation of the eigenvalue, perform full discretization using the OC method in the spatial direction and the backward Euler method in the time direction to obtain the backward Euler type high-order format of the spatio-temporal sequence equation, and / or, based on the spatio-temporal sequence equation of the eigenvalue, perform discretization using the OC method and the time midpoint method in the spatial direction and the time direction respectively to obtain the time midpoint type high-order format of the spatio-temporal sequence equation; A ground state solution module configured to, based on the target problem, perform iterative solution on the backward Euler type high-order format of the spatio-temporal sequence equation and / or the time midpoint type high-order format of the spatio-temporal sequence equation to determine the ground state; An excited state calculation module configured to calculate the excited state according to the determined ground state.

9. An electronic device, characterized in that, Including: A processor, and a memory communicatively connected to the processor; The memory stores computer execution instructions; The processor executes the computer execution instructions stored in the memory to implement the BEC ground state and excited state determination method according to any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores computer-executable instructions, which are used to implement the BEC ground state and excited state determination method according to any one of claims 1-7 when executed by a processor.

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