Schrodinger-based quantum preprocessing method

Abstract

The invention relates to a quantum preprocessing method based on Schrodinger, and the method comprises the steps: carrying out the finite element discrete BPX multi-level preprocessing of a to-be-solved partial differential equation, constructing a quantum block code corresponding to a preprocessing system, and obtaining a quantum preprocessing linear algebraic equation set; solving of a quantum preprocessing linear algebraic equation set is converted into steady-state solution solving of a linear ordinary differential equation set; schrodinger is adopted, and a linear ordinary differential equation set is converted into a Hamiltonian system capable of being calculated on a quantum computer through a dimension raising method; and designing a quantum circuit diagram for the Hamiltonian system by using a quantum Hamiltonian simulation algorithm, and simulating the quantum circuit diagram through a quantum computer to obtain a simulation result. The quantum preprocessing algorithm obtained through Schrodinger has the exponential acceleration advantage on the matrix order, precision and the like, the complexity influence caused by the condition number is eliminated, and the algorithm is simple and efficient.

Description

Technical Field

[0001] The present invention relates to the technical field of quantum computing applications, and in particular to a quantum preprocessing method based on Schrödinger transformation. Background Art

[0002] Numerical solutions to many problems in science and engineering face the dilemma of the curse of dimensionality, and classical computing methods are difficult to solve effectively. The core problem is the solution of linear algebraic equations. Quantum computers have natural parallelism and can achieve exponential acceleration in solving linear algebraic equations.

[0003] However, the optimal quantum algorithm still depends linearly on the condition number of the coefficient matrix, which makes the influence of the condition number still significant when dealing with large-scale problems. In particular, for mesh-dependent numerical algorithms (for example, finite element methods), the condition number of the coefficient matrix will grow rapidly as the mesh is encrypted, making quantum linear system algorithms face challenges brought by the condition number. Summary of the invention

[0004] The present invention provides a quantum preprocessing method based on Schrödingerization, which can eliminate the influence of the condition number in the time complexity of the problem obtained by discretizing partial differential equations, significantly reduce computing resources, and solve the limitations of current quantum linear system algorithm simulation.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is: In a first aspect, a quantum preprocessing method based on Schrödingerization is provided, comprising: The partial differential equation to be solved is subjected to finite element discretization BPX multi-level preprocessing, and the quantum block code corresponding to the preprocessing system is constructed to obtain a quantum preprocessing linear algebraic equation system; Transform the solution of quantum preconditioned linear algebraic equations into the solution of steady-state solutions of linear ordinary differential equations; Using Schrodinger transformation, the linear ordinary differential equations are transformed into a Hamiltonian system that can be calculated on a quantum computer using the dimensionality increase method; The quantum Hamiltonian simulation algorithm is used to design a quantum circuit diagram for the Hamiltonian system, and the quantum circuit diagram is simulated by a quantum computer to obtain simulation results.

[0006] Furthermore, the partial differential equation is Poisson's equation.

[0007] Furthermore, the partial differential equation to be solved is subjected to finite element discretization BPX multi-level preprocessing, and the quantum block code corresponding to the preprocessing system is constructed to obtain a quantum preprocessing linear algebraic equation group, including: Determine the Poisson equation to be solved including physical boundary conditions; Finite element discretization is performed on Poisson's equation to obtain a system of linear algebraic equations; A multi-level preprocessor of BPX multi-level preprocessor is obtained, a quantum block code is obtained according to the multi-level preprocessor, and a quantum preprocessing linear algebraic equation group is obtained.

[0008] Furthermore, the Poisson equation to be solved including the physical boundary conditions is determined, including: The expression of the Poisson equation to be solved is: ; in, is the calculation area; Neumann boundary, indicating the computational area on the right side; is the Dirichlet frontier; represents the Laplace operator, represents an unknown function; represents a known function; is the Dirichlet boundary condition function; is the Neumann boundary condition function; is the normal derivative of the region boundary.

[0009] Furthermore, the Poisson equation is discretized by finite element method to obtain a set of linear algebraic equations, including: Finite element discretization is performed on Poisson's equation, and the initial grid and refined grid are set as , forming a set of nested grids, and the linear algebraic equations on the last layer are ; is the number of layers; Corresponding to the initial grid; Corresponding encryption Second-rate; A coefficient matrix representing a linear equation system obtained by finite element discretization of the Poisson equation; Represents the right-hand side vector of the linear equation system obtained by finite element discretization of the Poisson equation; is the solution to the resulting linear equations.

[0010] Further, a multi-level preprocessor of BPX multi-level preprocessor is obtained, and quantum block coding is obtained according to the multi-level preprocessor to obtain a quantum preprocessing linear algebraic equation group, including: Get the multi-level preprocessing submodule of BPX multi-level preprocessing ;in, for The mesh size; Represents spatial dimension; For the The extended matrix from the layer grid to the last layer grid; To extend the matrix; for The Hermitian conjugate matrix of ; The quantum block code is obtained according to the multi-level preprocessor, and the quantum preprocessing linear algebraic equation group is obtained.

[0011] Furthermore, the expression of the linear ordinary differential equation system is: ; in, represents the solution of the ordinary differential equations corresponding to the preconditioned linear algebraic equations, express The derivation of time, ; ; ; ; for The Hermitian conjugate matrix of .

[0012] Furthermore, Schrödinger transformation is adopted to transform the linear ordinary differential equations into a Hamiltonian system that can be calculated on a quantum computer using the dimensionality increase method, including: By using Schrödinger transformation, auxiliary variables are added to the linear ordinary differential equation system to obtain the homogeneous linear ordinary differential equation system. The expression of the homogeneous linear ordinary differential equation system is: ; in, represents the solution of the homogenized linear ordinary differential equation system, represents the coefficient matrix of the homogenized linear ordinary differential equation system, ; ; is the evolution time required for the steady-state solution, is the identity matrix; Transform the homogeneous ordinary differential equations into convection equations. The expression of the convection equation is: ; ; in, represents the solution of the convection equation derived from the homogenized linear ordinary differential equation system, represents the imaginary unit, is the natural logarithm constant, is the auxiliary variable added, express The partial derivative of To discretize the value, The corresponding infinite region is truncated to the interval ; is a known function; For the The Hermitian conjugate matrix of ; and satisfy and , Representation Matrix The maximum absolute value of a negative eigenvalue; Representation Matrix The maximum absolute value of the positive eigenvalue; is the expected accuracy of the algorithm; The region is evenly divided into Get the grid nodes ; is the node sequence number; exist The direction is discretized by spectral method to obtain a discrete system, and the expression of the spatial discrete system is: ; in, represents the system function after the auxiliary variables are discretized, express The identity matrix of qubits, is a diagonal matrix, , ; is the quantum Fourier transform; is the inverse quantum Fourier transform; is the tensor product operator for matrices; Construct the Schrodinger-ized Hamiltonian system. The expression of the Hamiltonian system is: ; in, is a unitary transformation, , It indicates the range within which the Schrodingerization method can recover the original solution.

[0013] The beneficial effects achieved by the present invention are: The partial differential equation to be solved is subjected to finite element discretization BPX multi-level preprocessing, and the quantum block code corresponding to the preprocessing system is constructed to obtain the quantum preprocessing linear algebraic equation group; the solution of the quantum preprocessing linear algebraic equation group is converted into the steady-state solution of the linear ordinary differential equation group; Schrodinger transformation is adopted, and the linear ordinary differential equation group is transformed into a Hamiltonian system that can be calculated on a quantum computer using the dimension-raising method; the quantum Hamiltonian simulation algorithm is used to design the quantum circuit diagram for the Hamiltonian system, and the simulation results are obtained by simulating the quantum circuit diagram with a quantum computer. The quantum preprocessing algorithm obtained by Schrodinger transformation not only has exponential acceleration advantages in terms of matrix order and accuracy, but also eliminates the complexity caused by the condition number. The algorithm is simple and efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 It is a flow chart of the quantum preprocessing method based on Schrödinger transformation of the present invention; Figure 2 Schematic diagram of the calculation area and initial grid of the present invention; Figure 3 The block diagram of the Hamiltonian system after Schrodingerization of the present invention; Figure 4 It is a schematic diagram of the calculation area after uniform division of the present invention; Figure 5 This is a simulation result diagram output after Schrodinger transformation of the present invention; Figure 6 It is the accurate solution diagram output after Schrodinger transformation of the present invention; Figure 7 For the present invention T =15 error order diagram given by preprocessing; Figure 8 For the present invention T =15 is the error order diagram without preprocessing; Fig. 9 For the present invention T Schematic diagram of the error order without preprocessing when =40. DETAILED DESCRIPTION

[0015] The present invention will be further described below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and cannot be used to limit the protection scope of the present invention.

[0016] like Figure 1 As shown, an embodiment of the present invention provides a quantum preprocessing method based on Schrödinger transformation, comprising: 101, performing finite element discretization BPX multi-level preprocessing on the partial differential equation to be solved, constructing a quantum block code corresponding to the preprocessing system, and obtaining a quantum preprocessing linear algebraic equation group; In this embodiment, the partial differential equation specifically takes the Poisson equation as an example. In practical applications, the partial differential equation may also be other equations such as a biharmonic equation. The expression of the Poisson equation to be solved is: ; in, is the calculation area; It is the Neumann boundary; is the Dirichlet frontier; represents the Laplace operator; represents an unknown function; represents a known function; is the Dirichlet boundary condition function; is the Neumann boundary condition function; is the normal derivative of the region boundary; Specific calculation area , Take the calculation area On the right side of , the finite element space is taken as a nested first-order Lagrangian element, such as Figure 2 As shown, it is a schematic diagram of the calculation area and the initial grid; Finite element discretization is performed on Poisson's equation, and the initial grid and refined grid are set as , forming a set of nested grids, and the linear algebraic equations on the last layer are ; is the number of layers; Corresponding to the initial grid; Corresponding encryption Second-rate; Represents the coefficient matrix of the linear equation system obtained by finite element discretization of Poisson's equation; Represents the right-hand side vector of the linear equation system obtained by finite element discretization of Poisson's equation; is the solution of the obtained linear equation system;

[0017] Get the multi-level preprocessing submodule of BPX multi-level preprocessing ;in, for The mesh size; Represents spatial dimension; For the The extended matrix from the layer grid to the last layer grid; for The Hermitian conjugate matrix of ; BPX multi-level preconditioner (Bramble-Pasciak-Xu Preconditioner) is a preconditioning technology proposed by Xu Jinchao and others. It is mainly used to solve the influence of condition number on algorithm time complexity in numerical solution of partial differential equations. BPX multi-level preconditioner occupies an important position in existing multi-grid methods, especially in parallel computing, and is widely used in scientific computing. It is based on the idea of ​​multi-level (or multi-grid) method, and constructs multi-level preconditioner by combining operator information of different levels (coarse and fine grids), which significantly reduces the condition number of the system, thereby improving the convergence speed of the iterative method. After obtaining the multi-level preprocessing Afterwards, the quantum block code is obtained according to the multi-level preprocessor, and the quantum preprocessing linear algebraic equation system is obtained; Quantum block encoding is a coding strategy most commonly used in quantum computing. It is mainly used to embed a non-unitary matrix into a larger unitary matrix for efficient processing on a quantum computer.

[0018] 102, transform the solution of quantum preconditioned linear algebraic equations into the solution of steady-state solutions of linear ordinary differential equations; The expression of the linear ordinary differential equation system is: ; in, represents the solution of the ordinary differential equations corresponding to the preconditioned linear algebraic equations, express The derivation of time, ; ; ; ; for The Hermitian conjugate matrix of is the coefficient matrix of the linear algebraic equations, is the right-hand side vector in the linear algebraic equation system;

[0019] The calculated steady-state solution corresponds to the target variable.

[0020] 103, using Schrödinger transformation and dimension-raising method to transform a system of linear ordinary differential equations into a Hamiltonian system that can be calculated on a quantum computer; By using Schrödinger transformation, auxiliary variables are added to the linear ordinary differential equation system to obtain the homogeneous linear ordinary differential equation system. The expression of the homogeneous linear ordinary differential equation system is: ; in, represents the solution of the homogenized linear ordinary differential equation system, represents the coefficient matrix of the homogenized linear ordinary differential equation system, ; ; is the evolution time required for the steady-state solution, is the identity matrix; Transform the homogeneous ordinary differential equations into convection equations. The expression of the convection equation is: ; ; in, represents the solution of the convection equation derived from the homogenized linear ordinary differential equation system, represents the imaginary unit, is the natural logarithm constant, is the auxiliary variable added, express The partial derivative of To discretize the value, The corresponding infinite region is truncated to the interval ; is a known function; for The Hermitian conjugate matrix of ; and satisfy and , Representation Matrix The maximum absolute value of a negative eigenvalue; Representation Matrix The maximum absolute value of the positive eigenvalue; is the expected accuracy of the algorithm; The region is evenly divided into Get the grid nodes ; is the node sequence number; It should be noted that in this embodiment, the simulation time required for the steady-state solution is set to , The number of directions divided is ; The exact solution is ; exist p The direction is discretized by spectral method to obtain a discrete system, and the expression of the discrete system is: ; in, represents the system function after the auxiliary variables are discretized, express The identity matrix of qubits, is a diagonal matrix, , ; is the quantum Fourier transform; is the inverse quantum Fourier transform; is the tensor product operator for matrices; Construct the Schrodingerized Hamiltonian system. Figure 3 The block diagram of the Hamiltonian system after Schrödingerization is shown. Figure 3 middle ; ; is a unitary transformation; ; The expression of the Hamiltonian system is: ; Project or measure the solution to the computational ground state , indicates that the Schrodinger method can recover the range of the original solution, where k satisfy , and then obtain the target variable ; It should be noted that the quantum computer in this embodiment includes but is not limited to quantum computers using technologies such as superconducting quantum bits, ion trap quantum bits or topological quantum bits.

[0021] 104. Use the quantum Hamiltonian simulation algorithm to design a quantum circuit diagram for the Hamiltonian system, and simulate the quantum circuit diagram through a quantum computer to obtain the simulation result.

[0022] In this embodiment, after the Hamiltonian system is obtained in step 103, a quantum circuit diagram is designed for the Hamiltonian system using a quantum Hamiltonian simulation algorithm. Since a quantum circuit diagram can only be efficiently run in a quantum computer, a quantum computer needs to be used to simulate the quantum circuit diagram to obtain a simulation result. Figure 4 The figure shows the calculation area after uniform division; Figure 5 The figure shows the simulation result after Schrodinger transformation. Figure 6 This is the exact solution diagram output after Schrödinger transformation; by comparison Figure 5 and Figure 6 It can be seen that the simulation results match the exact solution well, proving that the simulation results are correct; like Figure 7 Shown T = 15, the preprocessing gives and The error orders are and , which is consistent with the theoretical expectation of finite element method. Figure 8 Shown T = 15, the error order without preprocessing is reduced. When the evolution time is increased toT =40, such as Fig. 9 Shown T =40, the error order is consistent with theoretical expectations; this shows that if no preprocessing is performed, more evolution time is required to make the error order consistent with theoretical expectations. Therefore, quantum preprocessing can reduce the evolution time and reduce the computational complexity.

[0023] The beneficial effects of the quantum preprocessing method based on Schrödingerization in the embodiment of the present invention are: The partial differential equation to be solved is subjected to finite element discretization BPX multi-level preprocessing, and the quantum block code corresponding to the preprocessing system is constructed to obtain the quantum preprocessing linear algebraic equation group; the solution of the quantum preprocessing linear algebraic equation group is converted into the steady-state solution of the linear ordinary differential equation group; Schrodinger transformation is adopted, and the linear ordinary differential equation group is transformed into a Hamiltonian system that can be calculated on a quantum computer using the dimension-raising method; the quantum Hamiltonian simulation algorithm is used to design the quantum circuit diagram for the Hamiltonian system, and the simulation results are obtained by simulating the quantum circuit diagram with a quantum computer. The quantum preprocessing algorithm obtained by Schrodinger transformation not only has exponential acceleration advantages in terms of matrix order and accuracy, but also eliminates the complexity caused by the condition number. The algorithm is simple and efficient.

[0024] In the above Figure 1 In the embodiment shown, step 101 describes the use of a multi-level preprocessor for quantum block coding. In the specific implementation process, considering that the multi-level preprocessor in the BPX multi-level preprocessing technology is a multi-level structure, when performing quantum block coding, the designed quantum block coding strategy uses a hierarchical quantum block coding strategy, which is as follows: (1), obtain the quantum block encoding of the prolongation matrix between all adjacent layers, as shown in i Layer and i The extension matrix between +1 layers is recorded as ; (2) Using the relation , obtain the j Layer to top layer J The extended matrix The quantum block encoding of Quantum block coding and preconditioning of linear systems.

[0025] The above steps (1) and (2) clearly illustrate the quantum block encoding process combined with multi-level preprocessors.

[0026] The linear system obtained by discretizing the partial differential equation and the corresponding multi-level preprocessor are quantum block encoded to obtain the preprocessed quantum linear system, and the solution of the system is converted into the solution of the steady-state solution of the linear ordinary differential equation; auxiliary variables are added to convert the non-homogeneous linear algebraic equation system after spatial discretization into a linear homogeneous ordinary differential equation system. The preprocessed system is transformed into a Hamiltonian system suitable for quantum simulation using the Schrodinger method. On this basis, the quantum circuit diagram of the Hamiltonian system generated by Schrodingerization is designed, and the data on the auxiliary bits are reasonably selected for measurement to obtain the target variable. The quantum preprocessing algorithm obtained by the present invention using Schrodingerization not only has exponential acceleration advantages in terms of matrix order and accuracy, but also eliminates the complexity influence caused by the condition number, and the algorithm is simple and efficient.

[0027] It will be appreciated by those skilled in the art that embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Furthermore, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0028] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0029] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0030] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process in the computer or other programmable device. Figure 1 A process or multiple processes and / or boxes Figure 1 A step that specifies a function in one or more boxes.

[0031] The above are merely embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention are included in the scope of the claims of the present invention to be approved.

Claims

1. A quantum preprocessing method based on Schrödinger transformation, characterized in that: include: The partial differential equation to be solved is subjected to finite element discretization BPX multi-level preprocessing, and the quantum block code corresponding to the preprocessing system is constructed to obtain a quantum preprocessing linear algebraic equation system; Converting the solution of the quantum preprocessing linear algebraic equations into a steady-state solution of a linear ordinary differential equation system; Using Schrodinger transformation and dimension-raising method, the linear ordinary differential equations are transformed into a Hamiltonian system that can be calculated on a quantum computer; A quantum circuit diagram is designed for the Hamiltonian system using a quantum Hamiltonian simulation algorithm, and a simulation result is obtained by simulating the quantum circuit diagram using the quantum computer.

2. The quantum preprocessing method based on Schrödinger transformation according to claim 1, characterized in that: The partial differential equation is Poisson's equation.

3. The quantum preprocessing method based on Schrödinger transformation according to claim 2, characterized in that: The partial differential equation to be solved is subjected to finite element discretization BPX multi-level preprocessing, and a quantum block code corresponding to the preprocessing system is constructed to obtain a quantum preprocessing linear algebraic equation group, including: Determine the Poisson equation to be solved including physical boundary conditions; Performing finite element discretization on the Poisson equation to obtain a set of linear algebraic equations; A multi-level preprocessor of BPX multi-level preprocessor is obtained, and a quantum block code is obtained according to the multi-level preprocessor to obtain a quantum preprocessing linear algebraic equation group.

4. The quantum preprocessing method based on Schrödinger transformation according to claim 3, characterized in that: The determining of the Poisson equation to be solved including the physical boundary conditions comprises: The expression of the Poisson equation to be solved is: ； Among them, the is the calculation area; is the Neumann boundary; is the Dirichlet boundary; represents the Laplace operator; Represents an unknown function; represents a known function; is the Dirichlet boundary condition function; is the Neumann boundary condition function; is the normal derivative of the region boundary.

5. The quantum preprocessing method based on Schrödinger transformation according to claim 4, characterized in that: The Poisson equation is subjected to finite element discretization to obtain a set of linear algebraic equations, including: The Poisson equation is discretized by finite element method, and the initial grid and the refined grid are set as , forming a set of nested grids, and the linear algebraic equations on the last layer are ; is the number of layers; Corresponding to the initial grid; Corresponding encryption times; represents the coefficient matrix of the linear equation system obtained by finite element discretization of the Poisson equation; represents the right-hand side vector of the linear equation system obtained by finite element discretization of the Poisson equation; is the solution to the resulting linear equations.

6. The quantum preprocessing method based on Schrödinger transformation according to claim 5, characterized in that: The step of obtaining a multi-level preprocessor for BPX multi-level preprocessing, obtaining a quantum block code according to the multi-level preprocessor, and obtaining a quantum preprocessing linear algebraic equation group includes: Get the multi-level preprocessing submodule of BPX multi-level preprocessing ; Among them, the For the The grid size of Represents the spatial dimension; For the The extended matrix of the layer grid to the last layer grid; For the The Hermitian conjugate matrix of ; The quantum block code is obtained according to the multi-level preprocessor, and a quantum preprocessing linear algebraic equation group is obtained.

7. The quantum preprocessing method based on Schrödinger transformation according to claim 6, characterized in that: The expression of the linear ordinary differential equation system is: ； Among them, the represents the solution of the ordinary differential equations corresponding to the preprocessed linear algebraic equations, Indicates the The derivative with respect to time, ; ; ; ; For the The Hermitian conjugate matrix of .

8. The quantum preprocessing method based on Schrödinger transformation according to claim 7, characterized in that: The Schrodinger transformation is adopted to transform the linear ordinary differential equation group into a Hamiltonian system that can be calculated on a quantum computer by using a dimensionality increase method, including: By using Schrödinger transformation, auxiliary variables are added to the linear ordinary differential equation system to obtain a homogeneous linear ordinary differential equation system, and the expression of the homogeneous linear ordinary differential equation system is: ； Among them, the represents the solution of the homogenized linear ordinary differential equation system, represents the coefficient matrix of the homogenized linear ordinary differential equation system, ; ; is the evolution time required for the steady-state solution, is the identity matrix; The homogeneous ordinary differential equations are transformed into convection equations, and the expression of the convection equations is: ； ； Among them, the represents the solution of the convection equation derived from the homogenized linear ordinary differential equation system, represents an imaginary unit, is the natural logarithm constant, is the added auxiliary variable, Indicates the The partial derivative of To discretize the value, The corresponding infinite region is truncated to the interval ; is a known function; For the The Hermitian conjugate matrix of ; Said and stated satisfy and , Represents the matrix The maximum absolute value of the negative eigenvalue; Represents the matrix The maximum absolute value of the positive eigenvalue; is the expected accuracy of the algorithm; The region is evenly divided into Get the grid nodes ; is the node sequence number; In the The direction is discretized by spectral method to obtain a discrete system, and the discrete system expression is: ； Among them, the represents the system function after the auxiliary variable is discretized, express The unit matrix of quantum bits is is a diagonal matrix, , ; is quantum Fourier transform; is the quantum inverse Fourier transform; is the tensor product operator for matrices; The Schrodinger-ized Hamiltonian system is constructed, and the expression of the Hamiltonian system is: ； Among them, the is a unitary transformation, , It indicates the range within which the Schrodingerization method can recover the original solution.

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