A high-efficiency simulation method for quantum system based on parallel reduction order
By employing parallel order reduction and time-parallel algorithms, the problem of low computational efficiency in high-dimensional quantum system simulation was solved, achieving efficient quantum system dynamics simulation, breaking through the time step limitation, and improving computational efficiency and accuracy.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- ANHUI UNIV
- Filing Date
- 2026-04-02
- Publication Date
- 2026-06-09
AI Technical Summary
Existing technologies are computationally inefficient when simulating high-dimensional quantum systems, and the memory and time costs are prohibitive. In particular, the number of grid points increases exponentially in high-dimensional scenarios, leading to a sharp increase in computational load, and matrix solving becomes the main computational burden.
A parallel reduction method is adopted, which projects the high-dimensional matrix of the Schrödinger equation into a low-dimensional subspace through Arnoldi iteration, and solves the equation using a time-parallel algorithm. This breaks through the limitation of the time step by the CFL condition, reduces computational complexity, and improves simulation efficiency.
It significantly improves the computational efficiency of high-dimensional quantum systems, maintains high accuracy and numerical stability, and provides a feasible way to simulate the dynamics of complex quantum systems over a long period of time.
Smart Images
Figure CN122175031A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of simulated quantum systems technology, specifically to an efficient simulation method for quantum systems based on parallel order reduction. Background Technology
[0002] The core task of quantum system simulation is essentially solving the Schrödinger equation, which describes its dynamics. Currently, various numerical methods have been developed for solving this equation, mainly including variational methods, finite difference methods, finite element methods, and emerging quantum variational algorithms. Among these, the finite difference method, due to its intuitive principle, simple mesh generation, and ease of algorithm implementation, exhibits good performance and accuracy in simulating simple low-dimensional quantum systems (such as one-dimensional potential wells and two-dimensional quantum dots). However, the computational efficiency of the finite difference method is severely limited by the system's dimensionality. When the simulated object expands to higher-dimensional space, the required number of mesh points increases exponentially, leading to a dramatic increase in computational cost. This significantly reduces the simulation efficiency of this method in high-dimensional scenarios, often making memory and time costs prohibitively high.
[0003] Generally, classical computers cannot handle the memory and computational demands of simulating high-dimensional quantum systems. For example, in 3-dimensional space, even taking only 100 points per dimension results in millions of grid points, corresponding to a Hamiltonian matrix of order in the millions. Direct diagonalization is virtually impossible, and pursuing higher accuracy exponentially increases computational costs. To complete the computation within available resources, coarser grids are often used, sacrificing solution accuracy, especially for regions with drastically changing wave functions (such as the edge of a potential well). For large-scale problems, matrix solving becomes the primary computational burden. The convergence speed of the system heavily depends on the condition number of the matrix and the chosen preconditioner, and designing efficient preconditioners for quantum problems is itself a challenge.
[0004] Based on this, the present invention provides an efficient simulation method for quantum systems based on parallel order reduction to solve the aforementioned technical problems. Summary of the Invention
[0005] The purpose of this invention is to provide an efficient simulation method for quantum systems based on parallel order reduction, thereby solving the problems mentioned in the background.
[0006] To achieve the above objectives, the present invention provides the following technical solution: This invention proposes an efficient simulation method for quantum systems based on parallel order reduction, comprising the following steps: S1. Receive the physical parameters and simulation requirements of the quantum system, wherein the physical parameters include electron mass, potential well size, and initial parameters of wave packets, and the simulation requirements include simulation physical time and accuracy requirements; S2. Based on the physical parameters, establish a time-dependent Schrödinger equation describing the dynamic behavior of the quantum system. The Schrödinger equation is rearranged into matrix form through the Kronecker product. The unitary property of the matrix is used to ensure the stability of the simulation solution. S3. The Arnoldi iteration is used to reduce the order of the matrix form of the Schrödinger equation by projecting the large matrix in the original space to a low-dimensional subspace to obtain a matrix of smaller order, thereby reducing computational complexity. S4. The reduced-order Schrödinger equation is solved using a time-parallel algorithm, which breaks through the limitation of time step size by the CFL condition and obtains a stable solution through fewer time steps of iteration; S5. Restore the solution of the reduced-order equation to the wave function of the quantum system, and complete the dynamic evolution simulation of the quantum system.
[0007] Preferably, the initial parameters of the wave packet in step S1 include the spatial broadening of the wave packet and the wave vectors propagating in the x, y, and z directions. The physical parameters also include the number of spatial grids, which corresponds to the number of grid divisions in the x, y, and z directions, and are denoted as follows: , , .
[0008] 3. In the efficient simulation method for quantum systems based on parallel order reduction according to claim 1 or 2, the time-dependent Schrödinger equation established in step S2 is shown in equation (1): ; in, For complex units, To reduce Planck's constant, m is the electron mass. Let be the wave function of the electron. The second derivative with respect to space in the above equation can be approximated by the second-order central difference. The second-order spatial second derivative is replaced by the second-order central difference approximation, and the replacement formula is shown in equation (2): ; in, , , To represent the number of spatial grids in different directions, and to facilitate the calculation of equations using matrices... Using the Kronecker product formula Organize; Substituting equation (2) back into equation (1) yields a matrix-formable equation, which, after transformation by the Kronecker product, forms an equation of order [order missing]. The large matrix A, and the Schrödinger equation in matrix form, are shown in equation (3): ; Where, matrix A is a To simplify the solution for a large matrix of order A, the Arnoldi iteration is used to reduce the order of matrix A. A smaller matrix is obtained by projecting it onto the feature space to replace the original large matrix for the solution.
[0009] Preferably, the order reduction process in step S3 is as follows: the orthogonal matrix Q and the Heisenberg matrix H are obtained through Arnoldi iteration, and the iteration relationship is shown in equation (4): ; Where Q is a The matrix H is of order 1. Both of these can be obtained by iterating through the Arnoldi formula. The equations can be The order of conversion ( The equation; Projecting the original matrix A onto the low-dimensional subspace generated by Q achieves the transformation of the original equation from higher to lower order. This is done through variable substitution, as shown in formulas (5) and (6): ; make ; The reduced-order equation is shown in equation (7): ; The order m of the reduced equation is much smaller than the order of the original matrix A. .
[0010] Preferably, in step S3, the order m of the reduced-order model is determined based on the number of spatial grids. , , It is necessary to strike a balance between computational accuracy and efficiency, avoiding either excessively high order which would result in limited efficiency gains or excessively low order which would lead to the loss of feature modes.
[0011] Preferably, the implementation process of the time-parallel algorithm in step S4 is as follows: the total simulation time domain is divided into multiple sub-intervals, corresponding to the total number of time iteration steps. The reduced-order first-order time-dependent differential equation is approximated and then integrated using the Toeplitz matrix, as shown in equation (8): ; in, Indicates an approximate relationship. This represents the total number of iterations in time. The system of equations is shown in equation (9): ; Based on the auxiliary definition, the recursive relationship between each time step is established, as shown in equations (10) and (11): ; ; Where b is obtained from the initial conditions of the wave function. Let the time steps be the time steps, and the relationships between the time steps satisfy a recursive relation: , It is a constant. for An identity matrix of order 1.
[0012] Preferably, in step S4, a non-uniform time step is used. Through variable substitution, the integrated matrix B is guaranteed to be diagonalizable, in the equation... Left multiplication of both ends Equation (12) is obtained: ; make The non-uniform step size used above ensures that matrix B can be diagonalized; Diagonalization relations are shown in equations (13) and (14):
[0013] ; Substituting the diagonalized result into formula (12) yields formula (15): ; For the formula By sequentially inverting each term in the equation, the results at each discrete time point can be obtained, which represent the projection of the wavefunction onto the reduced-dimensional subspace. Subsequently, by multiplying by the transpose of the Q matrix, the projection result can be mapped back to the original space to obtain the final complete wavefunction. By solving for the intermediate results at each time step, the solution process of the system of equations is simplified, as shown in equation (16): .
[0014] Preferably, the method further includes step S6: verifying the simulation results, including calculating the real part, imaginary part, and electron existence probability of the wave function, the square of the wave function modulus, performing error analysis based on the calculation results of the unconditional stability method, and verifying the accuracy of the simulation results; at the same time, performing global summation of the probability density of all grid points in the space to check the probability conservation characteristics and verify the numerical stability.
[0015] Preferably, the verification criterion for the probability conservation property is: the degree of deviation between the sum of the probabilities of the electrons in the whole space during the simulation and the initial value is within a preset range, wherein the preset range is close to 1 and the deviation is less than 1×10-7.
[0016] Preferably, the quantum system comprises a three-dimensional infinitely deep quantum potential well, the potential energy of which is expressed as in equation (17): ; in, It is the size of the potential well; The initial wave function of the electron is a Gaussian wave packet, expressed in equation (18): ; in, To broaden the space of the wave packet, , , These are the wave vectors propagating in the x, y, and z directions, respectively. The simulation physical time is set to 2 fs. The simulation scenario is the dynamic evolution of electron wave packets in a three-dimensional infinitely deep quantum potential well, including simulations of wave packet propagation, diffraction, and interference behavior.
[0017] Compared with the prior art, the beneficial effects of the present invention are: This invention decomposes long-term evolution processes into multiple concurrent computation segments through parallel decomposition in the time dimension, effectively utilizing multi-core and distributed computing resources. It also combines model reduction techniques based on eigenmodes and projection bases to perform low-dimensional subspace mapping on high-dimensional physical systems, significantly reducing model degrees of freedom and computational complexity. The synergistic effect of these two techniques significantly improves computational efficiency while maintaining high accuracy and numerical stability, providing a feasible approach for long-term dynamic simulation of complex quantum systems. In summary, this invention uses the Kronecker product to organize the established Schrödinger equation into matrix form, facilitating subsequent matrix operations for solving. The unitary nature of the matrix ensures the stability of the simulation solution. The Arnoldi algorithm is used to reduce the order of the matrix-form Schrödinger equation, projecting the large matrix from the original space to a relatively smaller subspace, thus improving the efficiency of the simulation solution. The reduced-order Schrödinger equation is solved using a time-parallel algorithm. This approach overcomes the time step limitation imposed by the CFL condition, ensuring a stable solution can be obtained with fewer time steps, and further improves the solution efficiency. Attached Figure Description
[0018] Figure 1 This is a flowchart of the procedure for this invention; Figure 2 This is a line graph showing the real and imaginary parts of the simulated wave function of this invention; Figure 3 This is a graph showing the probability density distribution of the detection points over time in the simulation of this invention; Figure 4 This is a schematic diagram comparing the errors of different methods based on the results of the unconditionally stable method according to the present invention. Detailed Implementation
[0019] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.
[0020] Please see Figures 1 to 4 This invention proposes an efficient simulation method for quantum systems based on parallel order reduction, comprising the following steps: S1. Receive the physical parameters and simulation requirements of the quantum system. The physical parameters include electron mass, potential well size, and initial parameters of wave packets. The simulation requirements include simulation physical time and accuracy requirements. S2. Based on physical parameters, a time-dependent Schrödinger equation describing the dynamic behavior of a quantum system is established. The Schrödinger equation is rearranged into matrix form through the Kronecker product, and the unitarity of the matrix is used to ensure the stability of the simulation solution. S3. The Arnoldi iteration is used to reduce the order of the matrix form of the Schrödinger equation by projecting the large matrix in the original space to a low-dimensional subspace to obtain a matrix of smaller order, thereby reducing computational complexity. S4. The reduced-order Schrödinger equation is solved using a time-parallel algorithm, which breaks through the limitation of time step size by the CFL condition and obtains a stable solution through fewer time steps of iteration; S5. Restore the solution of the reduced-order equation to the wave function of the quantum system, and complete the dynamic evolution simulation of the quantum system.
[0021] It should also be noted that the initial parameters of the wave packet in step S1 include the spatial broadening of the wave packet and the wave vectors propagating in the x, y, and z directions. The physical parameters also include the number of spatial grids, which corresponds to the number of grid divisions in the x, y, and z directions, denoted as […]. , , .
[0022] 3. According to claim 1 or 2, in a method for efficient simulation of quantum systems based on parallel order reduction, the time-dependent Schrödinger equation established in step S2 is shown in equation (1): ; in, For complex units, To reduce Planck's constant, m is the electron mass. Let be the wave function of the electron. The second derivative with respect to space in the above equation can be approximated by the second-order central difference. The second-order spatial second derivative is replaced by the second-order central difference approximation, and the replacement formula is shown in equation (2): ; in, , , To represent the number of spatial grids in different directions, and to facilitate the calculation of equations using matrices... Using the Kronecker product formula Organize; Substituting equation (2) back into equation (1) yields a matrix-formable equation, which, after transformation by the Kronecker product, forms an equation of order [order missing]. The large matrix A, and the Schrödinger equation in matrix form, are shown in equation (3): ; Where, matrix A is a To simplify the solution for a large matrix of order A, the Arnoldi iteration is used to reduce the order of matrix A. A smaller matrix is obtained by projecting it onto the feature space to replace the original large matrix for the solution.
[0023] It should also be noted that the order reduction process in step S3 is as follows: the orthogonal matrix Q and the Heisenberg matrix H are obtained through Arnoldi iteration, and the iteration relationship is shown in equation (4): ; Where Q is a The matrix H is of order 1. Both of these can be obtained by iterating through the Arnoldi formula. The equations can be The order of conversion ( The equation; Projecting the original matrix A onto the low-dimensional subspace generated by Q achieves the transformation of the original equation from higher to lower order. This is done through variable substitution, as shown in formulas (5) and (6): ; make ; The reduced-order equation is shown in equation (7): ; The order m of the reduced equation is much smaller than the order of the original matrix A. .
[0024] It should also be noted that in step S3, the order m of the model after order reduction is determined by the number of spatial grids. , , It is necessary to strike a balance between computational accuracy and efficiency, avoiding either excessively high order which would result in limited efficiency gains or excessively low order which would lead to the loss of feature modes.
[0025] It should also be noted that the implementation process of the time-parallel algorithm in step S4 is as follows: the total simulation time domain is divided into multiple sub-intervals, corresponding to the total number of time iteration steps. The reduced-order first-order time-dependent differential equation is approximated and then integrated using the Toeplitz matrix, as shown in equation (8): ; in, Indicates an approximate relationship. This represents the total number of iterations in time. The system of equations is shown in equation (9): ; Based on the auxiliary definition, the recursive relationship between each time step is established, as shown in equations (10) and (11): ; ; Where b is obtained from the initial conditions of the wave function. Let the time steps be the time steps, and the relationships between the time steps satisfy a recursive relation: , It is a constant. for An identity matrix of order 1.
[0026] It should also be noted that step S4 uses a non-uniform time step setting. Through variable substitution, it ensures that the integrated matrix B can be diagonalized, as shown in the equation. Left multiplication of both ends Equation (12) is obtained: ; make The non-uniform step size used above ensures that matrix B can be diagonalized; Diagonalization relations are shown in equations (13) and (14):
[0027] ; Substituting the diagonalized result into formula (12) yields formula (15): ; For the formula By sequentially inverting each term in the equation, the results at each discrete time point can be obtained, which represent the projection of the wavefunction onto the reduced-dimensional subspace. Subsequently, by multiplying by the transpose of the Q matrix, the projection result can be mapped back to the original space to obtain the final complete wavefunction. By solving for the intermediate results at each time step, the solution process of the system of equations is simplified, as shown in equation (16): .
[0028] It should also be noted that step S6 is included: verifying the simulation results, including calculating the real part, imaginary part, and electron existence probability of the wave function, the square of the wave function modulus, and performing error analysis based on the calculation results of the unconditional stability method to verify the accuracy of the simulation results; at the same time, the probability density of all grid points in the space is globally summed to check the probability conservation characteristics and verify the numerical stability.
[0029] It should also be noted that the verification criterion for the probability conservation property is: the deviation of the sum of the probabilities of the electron's existence in the entire space during the simulation from the initial value is within a preset range, which is close to 1 and the deviation is less than 1 × 10⁻⁶. -7 .
[0030] It should also be noted that the quantum system includes a three-dimensional infinitely deep quantum potential well, the potential energy of which is expressed in equation (17): ; in, It is the size of the potential well; The initial wave function of the electron is a Gaussian wave packet, expressed in equation (18): ; in, To broaden the space of the wave packet, , , These are the wave vectors propagating in the x, y, and z directions, respectively. The simulation physical time is set to 2 fs. The simulation scenario is the dynamic evolution of electron wave packets in a three-dimensional infinitely deep quantum potential well, including simulation of wave packet propagation, diffraction and interference behavior.
[0031] For examples, please refer to Figures 1 to 3 In practical applications, this invention is based on a parallel order reduction method for efficient simulation of quantum systems, specifically including the following steps: The dynamic behavior of quantum systems is usually described by the Schrödinger equation, so the essence of simulating quantum systems lies in solving this equation; To clearly illustrate the implementation and effects of this scheme, we will subsequently compare the dynamics of electron wave packets in a three-dimensional infinitely deep quantum potential well by using different methods to simulate classical examples in a quantum system. The evolution of simulated quantum systems can be governed by the time-dependent Schrödinger equation: ; in, For complex units, To reduce Planck's constant, m is the electron mass. Let be the wave function of the electron. The second derivative with respect to space in the above equation can be approximated by the second-order central difference: ; in, , , To represent the number of spatial grids in different directions, and to facilitate the calculation of equations using matrices... Using the Kronecker product formula After rearranging and substituting back into the equation, we get: ; Where, matrix A is a To simplify the solution for a large matrix of order A, the Arnoldi iteration is used to reduce the order of matrix A. A smaller matrix is obtained by projecting it onto the eigenspace, which is then used to replace the original large matrix for the solution. The process is as follows: ; Where Q is a The matrix H is of order 1. Both of these can be obtained by iterating through the Arnoldi formula. The equations can be The order of conversion ( The equation for ) is: ; make ; equation It becomes: ; By following the above process, the order of differential equations can be reduced. To ensure the completeness and accuracy of the solution, the order of the model after order reduction needs to be set reasonably according to the number of grids: if the order is too high, the improvement in computational efficiency after order reduction will be limited; if the order is too low, important feature modes will be missing, which will lead to the distortion of the solution results. Therefore, a balance needs to be struck between accuracy and efficiency, and an appropriate reduction order needs to be selected; Equation (9) is a first-order time-dependent differential equation, which can be solved using the traditional explicit FDTD method or the unconditional stability method. However, the explicit FDTD method is constrained by the CFL stability condition, and despite the low order of the equations, a large number of time steps are still required to ensure numerical stability. Although unconditionally stable methods theoretically allow arbitrary time steps and are not strictly constrained by stability conditions, this does not mean that the step size can be increased indefinitely without affecting computational accuracy. In fact, an excessively large time step can introduce significant time discretization errors, leading to distortion of the physical evolution process. Specifically, this manifests as phase calculation deviations, distortion of wave packet dynamics, and deviations of the calculated results of key physical quantities such as energy and momentum from their true values. To further improve solution efficiency while ensuring solution accuracy, a time-parallel method is adopted for the equation. Solve the problem; For equation Perform the following difference discretization to obtain: ; in, Indicates an approximate relationship. This represents the total number of iterations in time. Using the Toeplitz matrix to integrate the equations ,get: ; ; ; Where b is obtained from the initial conditions of the wave function. Let the time steps be the time steps, and the relationships between the time steps satisfy a recursive relation: , It is a constant. for An identity matrix of order 1; In the equation Left multiplication of both ends get: ; make The non-uniform step size used above ensures that matrix B can be diagonalized, that is: ; ; Formula Substitute into the formula get: ; For the formula By inverting each item in sequence, the result at each discrete time point can be obtained, which is the projection of the wave function onto the reduced-dimensional subspace. Subsequently, by multiplying by the transpose of the Q matrix, the projection result can be mapped back to the original space to obtain the final complete wave function, whose expression is as follows: ; At this point, the wave function in the Schrödinger equation can be obtained; The flowchart of the solution is as follows: Figure 1 As shown; To verify the effectiveness of this invention, the same quantum system was simulated using the method, the FDTD method, and the unconditional stability method for comparison, thus verifying the correctness and advantages of the algorithm. Suppose there exists a three-dimensional infinite potential well in space, whose potential energy is expressed as follows: ; in, It is the size of the potential well; Initially, the electron's wave function is a Gaussian wave packet as follows: ; in, To broaden the space of the wave packet, , , These are the wave vectors of the wave packet propagating in the x, y, and z directions, respectively. In the simulation calculation, the simulation physical time is set to 2fs, and the other parameter settings are shown in Table 1. The parameter settings for different methods are shown in Table 2. Table 1: Simulation Parameter Settings for Quantum Systems
[0032] Table 2: Parameters of different methods
[0033] Figure 2 The results show a comparison of the evolution of the real and imaginary parts of the wave function over time using three different numerical methods at the same observation location. Figure 3 This presents the curves showing the change of the electron presence probability (i.e., the squared modulus of the wave function) calculated by the three methods over time at the same location; To quantitatively assess the accuracy of the calculation, Figure 4Based on the calculation results of the unconditionally stable method, error analysis was performed on the FDTD method and the time-parallel order reduction method. The error curves show that the error between the FDTD method and the benchmark is consistently below -20dB, while the error between the time-parallel order reduction method and the benchmark is consistently below -60dB. This indicates that the latter has significantly improved computational accuracy and superior numerical reliability compared to the former. Table 4 Simulation times for different methods
[0034] Table 4 shows a comparison of the simulation time for the three methods; Data shows that the time-parallel order reduction method has significantly better simulation efficiency than the other two methods while ensuring the stability of the calculation results, and it is an order of magnitude better than the unconditionally stable method. Table 5 Analysis of the conservation properties of electron presence probability using different methods
[0035] To comprehensively evaluate the stability of different numerical methods, the probability density of all spatial grid points is globally summed at time 2fs during the simulation. The ability of each algorithm to maintain probability conservation is quantified by examining the deviation of this sum from the initial value, thereby directly reflecting its numerical stability. The analysis results show that the unconditionally stable method, due to its use of global implicit matrix solving, strictly maintains unitarity in time evolution, and its probability sum remains around 1 throughout the simulation process, demonstrating excellent numerical stability. In contrast, traditional explicit FDTD methods are limited by the inherent limitations of CFL conditions and explicit schemes. The sum of probabilities will show visible drift and decay over time, especially in long-term or large-time-step simulations where stability is significantly insufficient and it is difficult to strictly maintain the probability conservation properties of quantum systems. The time-parallel order reduction method achieves a good balance between the two: while inheriting the stable framework of the unconditional method, it breaks through the CFL condition. Through model dimensionality reduction and time-parallel optimization of the solution process, although it introduces slight numerical perturbations, it can still maintain a high level of probability conservation. Its stability is far better than the traditional FDTD method and close to that of the unconditional stable method.
[0036] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.
[0037] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.
Claims
1. An efficient simulation method for quantum systems based on parallel order reduction, characterized in that, Includes the following steps: S1. Receive the physical parameters and simulation requirements of the quantum system, wherein the physical parameters include electron mass, potential well size, and initial parameters of wave packets, and the simulation requirements include simulation physical time and accuracy requirements; S2. Based on the physical parameters, establish a time-dependent Schrödinger equation describing the dynamic behavior of the quantum system. The Schrödinger equation is rearranged into matrix form through the Kronecker product. The unitary property of the matrix is used to ensure the stability of the simulation solution. S3. The Arnoldi iteration is used to reduce the order of the matrix form of the Schrödinger equation by projecting the large matrix in the original space to a low-dimensional subspace to obtain a matrix of smaller order, thereby reducing computational complexity. S4. The reduced-order Schrödinger equation is solved using a time-parallel algorithm, which breaks through the limitation of time step size by the CFL condition and obtains a stable solution through fewer time steps of iteration; S5. Restore the solution of the reduced-order equation to the wave function of the quantum system, and complete the dynamic evolution simulation of the quantum system.
2. The efficient simulation method for quantum systems based on parallel order reduction according to claim 1, characterized in that, The initial parameters of the wave packet in step S1 include the spatial broadening of the wave packet and the wave vectors propagating in the x, y, and z directions. The physical parameters also include the number of spatial grids, which corresponds to the number of grid divisions in the x, y, and z directions, and are denoted as follows: , , .
3. The efficient simulation method for quantum systems based on parallel order reduction according to claim 1 or 2, characterized in that, In step S2, the time-dependent Schrödinger equation is established, as shown in equation (1): ； in, For complex units, To reduce Planck's constant, m is the electron mass. Let be the wave function of the electron. The second derivative with respect to space in the above equation can be approximated by the second-order central difference. The second-order spatial second derivative is replaced by the second-order central difference approximation, and the replacement formula is shown in equation (2): （2）； in, , , To accommodate the number of spatial grids in different directions, the formula is simplified using the Kronecker product to facilitate matrix calculations. Substituting equation (2) back into equation (1) yields a matrix-formable equation, which, after transformation by the Kronecker product, forms an equation of order [order missing]. The large matrix A, and the Schrödinger equation in matrix form, are shown in equation (3): （3）； Where, matrix A is a To simplify the solution for a large matrix of order A, the Arnoldi iteration is used to reduce the order of matrix A. By projecting the matrix onto the feature space, a smaller matrix is obtained to replace the original large matrix for the solution.
4. The efficient simulation method for quantum systems based on parallel order reduction according to claim 3, characterized in that, The order reduction process in step S3 is as follows: the orthogonal matrix Q and the Heisenberg matrix H are obtained through Arnoldi iteration, and the iteration relationship is shown in equation (4): （4）； Where Q is a The matrix H is of order 1. A Heisenberg matrix, both of which can be obtained by Arnoldi iteration, and the order of the equation can be converted to a formula. ( The equation; Projecting the original matrix A onto the low-dimensional subspace generated by Q achieves the transformation of the original equation from higher to lower order. This is done through variable substitution, as shown in formulas (5) and (6): ； make ； The reduced-order equation is shown in equation (7): ； The order m of the reduced equation is much smaller than the order of the original matrix A. .
5. The efficient simulation method for quantum systems based on parallel order reduction according to claim 4, characterized in that, In step S3, the order m of the reduced model is determined based on the number of spatial grids. , , It is necessary to strike a balance between computational accuracy and efficiency, avoiding either excessively high order which would result in limited efficiency gains or excessively low order which would lead to the loss of feature modes.
6. A method for efficient simulation of quantum systems based on parallel order reduction according to any one of claims 1-5, characterized in that, The implementation process of the time-parallel algorithm in step S4 is as follows: the total simulation time domain is divided into multiple sub-intervals, corresponding to the total number of time iteration steps. The reduced-order first-order time-dependent differential equation is approximated and then integrated using the Toeplitz matrix, as shown in equation (8): ； in, Indicates an approximate relationship. This represents the total number of iterations in time. The system of equations is shown in equation (9): ； Based on the auxiliary definition, the recursive relationship between each time step is established, as shown in equations (10) and (11): ； ； Where b is obtained from the initial conditions of the wave function. Let the time steps be the time steps, and the relationships between the time steps satisfy a recursive relation: , It is a constant. for An identity matrix of order 1.
7. The efficient simulation method for quantum systems based on parallel order reduction according to claim 6, characterized in that, In step S4, a non-uniform time step is used. Through variable substitution, it is ensured that the integrated matrix B can be diagonalized, as shown in the equation. Left multiplication of both ends Equation (12) is obtained: ； make The non-uniform step size used above ensures that matrix B can be diagonalized; Diagonalization relations are shown in equations (13) and (14): ； ； Substituting the diagonalized result into formula (12) yields formula (15): ； For the formula By sequentially inverting each term in the equation, the results at each discrete time point can be obtained, which represent the projection of the wavefunction onto the reduced-dimensional subspace. Subsequently, by multiplying by the transpose of the Q matrix, the projection result can be mapped back to the original space to obtain the final complete wavefunction. By solving for the intermediate results at each time step, the solution process of the system of equations is simplified, as shown in equation (16): 。 8. A method for efficient simulation of quantum systems based on parallel order reduction according to any one of claims 1-7, characterized in that, The method also includes step S6: verifying the simulation results, including calculating the real part, imaginary part, and electron existence probability of the wave function, the square of the wave function modulus, and performing error analysis based on the calculation results of the unconditional stability method to verify the accuracy of the simulation results; at the same time, the probability density of all grid points in the space is globally summed to check the probability conservation characteristics and verify the numerical stability.
9. The efficient simulation method for quantum systems based on parallel order reduction according to claim 8, characterized in that, The verification criterion for the probability conservation property is: the degree of deviation between the sum of the probability of the electron in the whole space during the simulation and the initial value is within a preset range, wherein the preset range is close to 1 and the deviation is less than 1×10-7.
10. A method for efficient simulation of quantum systems based on parallel order reduction according to any one of claims 1-9, characterized in that, The quantum system comprises a three-dimensional infinitely deep quantum potential well, the potential energy of which is expressed as in equation (17): ； in, It is the size of the potential well; The initial wave function of the electron is a Gaussian wave packet, expressed in equation (18): ； in, To broaden the space of the wave packet, , , These are the wave vectors propagating in the x, y, and z directions, respectively. The simulation physical time is set to 2 fs. The simulation scenario is the dynamic evolution of electron wave packets in a three-dimensional infinitely deep quantum potential well, including simulations of wave packet propagation, diffraction, and interference behavior.