A quantum preprocessing method based on Schrödingerization

Abstract

The present invention relates to a quantum preprocessing method based on Schrödingerization. The method includes: performing finite element discretization of the BPX multilevel preprocessing on the partial differential equation to be solved, constructing a quantum block encoding corresponding to the preprocessing system to obtain a quantum preprocessed linear algebraic equation set; transforming the solution of the quantum preprocessed linear algebraic equation set into the solution of the steady state of a linear ordinary differential equation set; adopting Schrödingerization, and using the dimension elevation method to transform the linear ordinary differential equation set into a Hamiltonian system that can be calculated on a quantum computer; designing a quantum circuit diagram for the Hamiltonian system by using a quantum Hamiltonian simulation algorithm, and obtaining a simulation result by simulating the quantum circuit diagram on a quantum computer. The quantum preprocessing algorithm obtained by using Schrödingerization not only has an exponential acceleration advantage in terms of matrix order and accuracy, etc., but also eliminates the complexity influence brought by the condition number, 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 Schrödingerization. Background Art

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

[0003] However, the optimal quantum algorithm still linearly depends on the condition number of the coefficient matrix, so that when dealing with large-scale problems, the influence of the condition number is still significant. Especially for grid-dependent numerical algorithms (e.g., the finite element method), as the grid is refined, the condition number of the coefficient matrix will increase rapidly, thus making the quantum linear system algorithm face the 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 for problems obtained by discretizing partial differential equations, significantly reduce computing resources, and solve the limitations of current quantum linear system algorithm simulations.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0006] In a first aspect, a quantum preprocessing method based on Schrödingerization is provided, including:

[0007] Performing BPX multilevel preprocessing of finite element discretization on the partial differential equation to be solved, constructing a quantum block encoding corresponding to the preprocessing system, and obtaining a quantum preprocessing system of linear algebraic equations;

[0008] Converting the solution of the quantum preprocessing system of linear algebraic equations into the solution of the steady state of a system of linear ordinary differential equations;

[0009] Adopting Schrödingerization and using the dimension-raising method to transform the system of linear ordinary differential equations into a Hamiltonian system that can be calculated on a quantum computer;

[0010] Designing a quantum circuit diagram for the Hamiltonian system by using the quantum Hamiltonian simulation algorithm, and obtaining a simulation result by simulating the quantum circuit diagram on a quantum computer.

[0011] Further, the partial differential equation is a Poisson equation.

[0012] Further, performing BPX multilevel preprocessing of finite element discretization on the partial differential equation to be solved, constructing a quantum block encoding corresponding to the preprocessing system, and obtaining a quantum preprocessing system of linear algebraic equations, including:

[0013] Determine the Poisson equation to be solved that includes physical boundary conditions;

[0014] Perform finite element discretization on the Poisson equation to obtain a system of linear algebraic equations;

[0015] Obtain a multi-level preconditioner for BPX multi-level preconditioning, obtain a quantum block encoding based on the multi-level preconditioner, and obtain a quantum preconditioned system of linear algebraic equations.

[0016] Furthermore, determining the Poisson equation to be solved that includes physical boundary conditions includes:

[0017] The expression of the Poisson equation to be solved is:

[0018] ;

[0019] where, is the computational domain; is the Neumann boundary, representing the right side of the computational domain ; is the Dirichlet boundary; represents the Laplace operator, represents the unknown function; represents the known function; is the Dirichlet boundary condition function; is the Neumann boundary condition function; is the normal derivative of the region boundary.

[0020] Furthermore, performing finite element discretization on the Poisson equation to obtain a system of linear algebraic equations includes:

[0021] Perform finite element discretization on the Poisson equation, set the initial grid and the refined grid as , form a set of nested grids, and obtain the system of linear algebraic equations on the last layer as ; is the number of layers; corresponds to the initial grid; corresponds to the refined times; represents the coefficient matrix of the system of linear equations obtained by finite element discretization of the Poisson equation; represents the right-hand vector of the system of linear equations obtained by finite element discretization of the Poisson equation; is the solution of the obtained system of linear equations.

[0022] Further, obtain the multilevel preprocessing sub - unit for BPX multilevel preprocessing, and obtain the quantum block encoding based on the multilevel preprocessing sub - unit to obtain the quantum preprocessed linear algebraic equations, including:

[0023] Obtain the multilevel preprocessing sub - unit for BPX multilevel preprocessing ; where is the grid size of; represents the spatial dimension; is the prolongation matrix from the -th layer grid to the last layer grid; is the prolongation matrix; is the Hermitian conjugate matrix of;

[0024] Obtain the quantum block encoding based on the multilevel preprocessing sub - unit to obtain the quantum preprocessed linear algebraic equations.

[0025] Further, the expression of the linear ordinary differential equations is:

[0026] ;

[0027] where represents the solution of the ordinary differential equations corresponding to the preprocessed linear algebraic equations, represents the derivative with respect to time, ; ; ; ; is the Hermitian conjugate matrix of.

[0028] Further, adopt Schrödingerization, and use the dimension - raising method to transform the linear ordinary differential equations into a Hamiltonian system that can be calculated on a quantum computer, including:

[0029] Adopt Schrödingerization, add auxiliary variables to the linear ordinary differential equations to obtain homogeneous linear ordinary differential equations, and the expression of the homogeneous linear ordinary differential equations is:

[0030] ;

[0031] where represents the solution of the homogeneous linear ordinary differential equations after homogenization, represents the coefficient matrix of the homogeneous linear ordinary differential equations after homogenization, ; ; is the evolution time required for the steady - state solution, is the identity matrix;

[0032] Transform the homogeneous ordinary differential equation system into a convection equation, and the expression of the convection equation is:

[0033] ;

[0034] ;

[0035] where, 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 added auxiliary variable, represents the partial derivative of, discretize numerically, and truncate the infinite region corresponding to to the interval ; is a known function; the is the Hermitian conjugate matrix of the ;

[0036] and satisfy and , represents the maximum absolute value of the negative eigenvalues of the matrix ; represents the maximum absolute value of the positive eigenvalues of the matrix ; is the expected accuracy of the algorithm; evenly divide the region into parts to obtain the grid nodes ; is the node number;

[0037] in the direction is discretized using the spectral method to obtain a discrete system, and the expression of the spatial discrete system is:

[0038] ;

[0039] where, represents the system function after discretization of the auxiliary variable, represents 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 of matrices;

[0040] Construct the Hamiltonian system after Schrödingerization. The expression of the Hamiltonian system is:

[0041] ;

[0042] where is a unitary transformation, , represents the range within which the Schrödingerization method can recover the original solution.

[0043] The beneficial effects achieved by the present invention are as follows:

[0044] Perform finite element discretization of the BPX multilevel preconditioning on the partial differential equation to be solved, construct the quantum block encoding corresponding to the preconditioned system, and obtain the quantum preconditioned linear algebraic equations; transform the solution of the quantum preconditioned linear algebraic equations into the solution of the steady state of a system of linear ordinary differential equations; adopt Schrödingerization, and use the method of increasing dimensions to transform the system of linear ordinary differential equations into a Hamiltonian system that can be calculated on a quantum computer; design a quantum circuit diagram for the Hamiltonian system using the quantum Hamiltonian simulation algorithm, and obtain the simulation result by simulating the quantum circuit diagram on a quantum computer. The quantum preconditioning algorithm obtained by using Schrödingerization not only has an exponential acceleration advantage in terms of matrix order and accuracy, etc., but also eliminates the complexity impact brought by the condition number, and the algorithm is simple and efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 is the flowchart of the quantum preconditioning method based on Schrödingerization of the present invention;

[0046] Figure 2 is the schematic diagram of the calculation region and the initial grid of the present invention;

[0047] Figure 3 is the block diagram of the Hamiltonian system after Schrödingerization of the present invention;

[0048] Figure 4 is the schematic diagram of the calculation region after uniform meshing of the present invention;

[0049] Figure 5 is the simulation result diagram output after Schrödingerization of the present invention;

[0050] Figure 6 is the exact solution diagram output after Schrödingerization of the present invention;

[0051] Figure 7 is for the present invention T = 15, the schematic diagram of the error order given by the preconditioning;

[0052] Figure 8 is for the present invention T = 15, the schematic diagram of the error order given without preconditioning;

[0053] Figure 9 For the present invention T Schematic diagram of the error order without preprocessing when = 40. Specific implementation manner

[0054] The present invention will be further described below with reference to 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.

[0055] As Figure 1 shown, an embodiment of the present invention provides a quantum preprocessing method based on Schrödingerization, including:

[0056] 101. Perform finite element discretized BPX multilevel preprocessing on the partial differential equation to be solved, construct a quantum block encoding corresponding to the preprocessing system, and obtain a quantum preprocessing linear algebraic equation system;

[0057] In this embodiment, the partial differential equation is specifically exemplified by the Poisson equation. In practical applications, the partial differential equation can also be other equations such as the biharmonic equation;

[0058] The expression of the Poisson equation to be solved is:

[0059] ;

[0060] Wherein, is the computational domain; is the Neumann boundary; is the Dirichlet boundary; represents the Laplace operator; represents the unknown function; represents the known function; is the Dirichlet boundary condition function; is the Neumann boundary condition function; is the normal derivative of the region boundary;

[0061] Specific computational domain , is taken as the right side of the computational domain , and the finite element space is taken as the nested first-order Lagrangian element. As Figure 2 shown, it is a schematic diagram of the computational domain and the initial grid;

[0062] Perform finite element discretization on the Poisson equation, set the initial grid and the refined grid as , form a set of nested grids, and obtain the linear algebraic equation system on the last layer as ; is the number of layers; corresponds to the initial grid; Corresponding encryption times; represents the coefficient matrix of the linear equations obtained by the finite element discretization of the Poisson equation; represents the right-hand vector of the linear equations obtained by the finite element discretization of the Poisson equation; is the solution of the obtained linear equations;

[0063] Obtain the multilevel preconditioner of BPX multilevel preconditioning ; where, is the mesh size of; represents the spatial dimension; is the prolongation matrix from the -th layer of mesh to the last layer of mesh; is Hermitian conjugate matrix of;

[0064] BPX multilevel preconditioning (Bramble-Pasciak-Xu Preconditioner) is a preconditioning technique proposed by Jinchao Xu et al., mainly used to solve the influence of the condition number on the algorithm time complexity in the numerical solution of partial differential equations. The BPX multilevel preconditioner plays an important role in the existing multigrid methods, especially in parallel computing, and is widely used in the scientific computing community; it is based on the idea of multilevel (or multigrid) methods, constructs a multilevel preconditioner by combining operator information at different levels (coarse and fine meshes), significantly reduces the condition number of the system, and thus improves the convergence speed of the iterative method;‌

[0065] After obtaining the multilevel preconditioner , obtain the quantum block encoding according to the multilevel preconditioner, and obtain the quantum preconditioned linear algebraic equations;

[0066] Quantum block encoding (Block-encoding)‌ is a most commonly used encoding strategy in quantum computing, mainly used to embed a non-unitary matrix into a larger unitary matrix for efficient processing on a quantum computer.

[0067] 102, transform the solution of the quantum preconditioned linear algebraic equations into the solution of the steady state of a linear ordinary differential equation;

[0068] The expression of the linear ordinary differential equation is:

[0069] ;

[0070] where, represents the solution of the ordinary differential equation corresponding to the preconditioned linear algebraic equations, represents Derivative with respect to time, ; ; ; ; is the Hermitian conjugate matrix of, is the coefficient matrix of the system of linear algebraic equations, is the right - hand vector in the system of linear algebraic equations;

[0071] The calculated steady - state solution corresponds to the target variable.

[0072] 103. Use Schrödingerization and the dimension - raising method to transform the system of linear ordinary differential equations into a Hamiltonian system that can be calculated on a quantum computer;

[0073] Use Schrödingerization, add auxiliary variables to the system of linear ordinary differential equations to obtain a homogeneous system of linear ordinary differential equations. The expression of the homogeneous system of linear ordinary differential equations is:

[0074] ;

[0075] where, represents the solution of the homogeneous system of linear ordinary differential equations after homogenization, represents the coefficient matrix of the homogeneous system of linear ordinary differential equations after homogenization, ; ; is the evolution time required for the steady - state solution, is the identity matrix;

[0076] Transform the homogeneous ordinary differential equations into a convection equation. The expression of the convection equation is:

[0077] ;

[0078] ;

[0079] where, represents the solution of the convection equation derived from the homogeneous system of linear ordinary differential equations after homogenization, represents the imaginary unit, is the natural logarithm constant, is the added auxiliary variable, represents the partial derivative of. Discretize numerically, and truncate the infinite region corresponding to to the interval ; is a known function; is the Hermitian conjugate matrix of;

[0080] and Satisfy and , denotes the maximum absolute value of the negative eigenvalues of matrix ; denotes the maximum absolute value of the positive eigenvalues of matrix ; is the desired accuracy of the algorithm; The region is evenly divided into parts to obtain the grid nodes ;

[0081] It should be noted that in this embodiment, the simulation duration required for the steady-state solution is set to , The number of parts divided in the direction is

[0082] The exact solution is taken as ;

[0083] In the p direction, the spectral method is used for discretization to obtain a discrete system, and the expression of the discrete system is:

[0084] ;

[0085] Among them, denotes the system function after discretization of the auxiliary variable, denotes the identity matrix of is a diagonal matrix, , ; is the quantum Fourier transform; is the inverse quantum Fourier transform; is the tensor product operator of matrices;

[0086] Construct the Hamiltonian system after Schrödingerization, Figure 3 The block diagram of the Hamiltonian system after Schrödingerization is shown in Figure 3 where ; ; is a unitary transformation; ; The expression of the Hamiltonian system is:

[0087] ;

[0088] Project or measure the solution onto the computational basis state , denotes the range within which the Schrödingerization method can recover the original solution, where the k satisfies , and then obtain the target variable ;

[0089] It should be noted that in this embodiment, the quantum computer includes, but is not limited to, quantum computers using technologies such as superconducting qubits, ion trap qubits, or topological qubits.

[0090] 104. Use the quantum Hamiltonian simulation algorithm to design a quantum circuit diagram for the Hamiltonian system, and obtain a simulation result by simulating the quantum circuit diagram with a quantum computer.

[0091] In this embodiment, after obtaining the Hamiltonian system in step 103, use the quantum Hamiltonian simulation algorithm to design a quantum circuit diagram for the Hamiltonian system. Since the quantum circuit diagram can only run efficiently in a quantum computer, a quantum computer needs to be used to simulate the quantum circuit diagram to obtain a simulation result; Figure 4 Shown is the computational domain after uniform dissection; Figure 5 Shown is the simulation result diagram output after Schrödingerization; Figure 6 Is the exact solution diagram output after Schrödingerization; By comparing Figure 5 and Figure 6 it can be seen that the simulation result has a good match with the exact solution, proving that the simulation result is correct;

[0092] Such as Figure 7 shown as T =15, the error orders of and given by the preprocessing are and respectively, which is consistent with the theoretical expectation of the finite element. Figure 8 Shown as T =15, the error order given without preprocessing is reduced. When the evolution time is increased to T =40, as Figure 9 shown as T =40, the error order is consistent with the theoretical expectation; thus it can be seen that if no preprocessing is performed, more evolution time is required to make the error order consistent with the theoretical expectation. Therefore, quantum preprocessing can reduce the evolution time and the computational complexity.

[0093] The beneficial effects of the quantum preprocessing method based on Schrödingerization in the embodiments of the present invention are:

[0094] Perform the finite element discretization of the partial differential equation to be solved by the BPX multilevel preconditioner, construct the quantum block encoding corresponding to the preconditioned system, and obtain the quantum preconditioned linear algebraic equations; transform the solution of the quantum preconditioned linear algebraic equations into the solution of the steady state of a linear ordinary differential equation; use the Schrödingerization method to transform the linear ordinary differential equation into a Hamiltonian system that can be calculated on a quantum computer by means of dimension elevation; design a quantum circuit diagram for the Hamiltonian system using the quantum Hamiltonian simulation algorithm, and obtain the simulation results by simulating the quantum circuit diagram on a quantum computer. The quantum preconditioning algorithm obtained by using the Schrödingerization method not only has an exponential acceleration advantage in terms of matrix order and accuracy, etc., but also eliminates the complexity impact brought by the condition number, and the algorithm is simple and efficient.

[0095] In the above Figure 1 In the embodiment shown, in step 101, it is described that the quantum block encoding is carried out by using the multilevel preconditioner. In the specific implementation process, considering that the multilevel preconditioner in the BPX multilevel preconditioning technology is a multi-layer structure, then when performing the quantum block encoding, the designed quantum block encoding strategy uses the hierarchical quantum block encoding strategy, which is specifically as follows:

[0096] (1), Obtain the quantum block encoding of the prolongation matrix between all adjacent layers. For example, the prolongation matrix between the i th layer and the i +1th layer is denoted as ;

[0097] (2), Use the relationship to obtain the quantum block encoding of the prolongation matrix j from the J th layer to the highest layer , and then obtain the quantum block encoding of and the preconditioned linear system.

[0098] The above steps (1) and (2) clearly explain the quantum block encoding process in combination with the multilevel preconditioner.

[0099] By performing quantum block encoding on the linear system obtained by discretizing partial differential equations and the corresponding multilevel preconditioner, a preconditioned quantum linear system is obtained, and the solution of the system is transformed into the solution of the steady state of a linear ordinary differential equation; auxiliary variables are added to transform the inhomogeneous linear algebraic equations after spatial discretization into linear homogeneous ordinary differential equations. Using the Schrödingerization method, the preconditioned system is transformed into a Hamiltonian system suitable for quantum simulation. On this basis, a quantum circuit diagram of the Hamiltonian system generated by Schrödingerization is designed, and the data on the auxiliary qubits are reasonably selected for measurement to obtain the target variables. The quantum preconditioning algorithm obtained by Schrödingerization in the present invention not only has exponential acceleration advantages in terms of matrix order and accuracy, etc., but also eliminates the complexity influence brought by the condition number, and the algorithm is simple and efficient.

[0100] Those skilled in the art should understand that the embodiments of the present invention can be provided as a method, a system, or a computer program product. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present invention can 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.

[0101] The present invention is described with reference to the 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, and the combination of processes and / or blocks in the flowchart and / or block diagram can be realized by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing devices generate a machine for realizing the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 the device specified in one block or multiple blocks.

[0102] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer-readable memory generate a manufactured article including an instruction device, and the instruction device realizes the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 the device specified in one block or multiple blocks.

[0103] These computer program instructions can also be loaded onto a computer or other programmable data processing apparatus, so that a series of operation steps are executed on the computer or other programmable apparatus to produce a computer-implemented process, thereby the instructions executed on the computer or other programmable apparatus provide steps for realizing the functions specified in one process or a plurality of processes and / or blocks Figure 1 one process or a plurality of processes and / or blocks Figure 1 and steps of the functions specified in one block or a plurality of blocks.

[0104] The above are only embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention are included in the scope of the claims of the present invention pending approval of the application.

Claims

1. A quantum preprocessing method based on Schrödingerization, characterized in that, Including: Determine the Poisson equation to be solved that includes physical boundary conditions; Perform finite element discretization on the Poisson equation, and set the initial mesh and the refined mesh as , forming a set of nested meshes, and obtaining the linear algebraic equations on the last layer as ; the is the number of layers; the corresponds to the initial mesh; the corresponds to the refined times; the represents the coefficient matrix of the linear algebraic equations; the represents the right-hand side vector of the linear algebraic equations; the is the solution of the linear algebraic equations; Obtain a multilevel preconditioner for BPX multilevel preconditioning, obtain a quantum block encoding based on the multilevel preconditioner, and obtain a quantum preconditioned linear algebraic system of equations; Transform the solution of the quantum preconditioned linear algebraic system of equations into the solution of the steady state of a linear ordinary differential equation system; Adopt Schrödingerization, use the dimension elevation method to transform the linear ordinary differential equation system into a Hamiltonian system that can be computed on a quantum computer; design a quantum circuit diagram for the Hamiltonian system using the quantum Hamiltonian simulation algorithm, and simulate the quantum circuit diagram through the quantum computer to obtain a simulation result; The expression of the linear ordinary differential equation system is: ； Among them, the represents the solution of the ordinary differential equation system corresponding to the preprocessed linear algebraic equation system, and the represents the derivative with respect to time. The ; the ; the ; the ; the is the Hermitian conjugate matrix of the ; the is the grid size of the ; the represents the spatial dimension; the is the extension matrix from the -th layer grid to the last layer grid.

2. The quantum preprocessing method based on Schrödingerization according to claim 1, wherein The obtaining of a multilevel preconditioner for BPX multilevel preconditioning, obtaining a quantum block encoding based on the multilevel preconditioner, and obtaining a quantum preconditioned linear algebraic system of equations includes: Obtain the multi-level preprocessing sub of BPX multi-level preprocessing ; The is the Hermitian conjugate matrix of the ; Obtain a quantum block encoding based on the multilevel preconditioner, and obtain a quantum preconditioned linear algebraic system of equations.

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