Electromagnetic simulation method based on Schrödingerization

Abstract

An electromagnetic simulation method in a homogeneous medium based on Schrödingerization. According to the Maxwell equations in a homogeneous medium, an equivalent matrix representation form is obtained; according to the classical spatial discretization format, the Yee format, complex physical boundary conditions are processed to obtain an ODE system; according to Schrödingerization, using the dimension-raising method, a Hamiltonian system that can be calculated on a quantum computer is obtained; according to the Lie-Trotter-Suzuki method, a quantum circuit diagram is constructed for the Hamiltonian system to record the process of applying a series of quantum operators to a quantum state in a graphical and compact manner. The present invention can significantly reduce computing resources and solves the limitations of current quantum algorithm simulations.

Description

Technical Field

[0001] The present invention relates to a technology in the field of quantum computing applications, specifically an electromagnetic simulation method in a homogeneous medium based on Schrödingerization. Background Art

[0002] The 5G era faces exponentially growing demands such as signal processing and wireless optical communication. Traditional algorithms such as the traditional FDTD algorithm or the finite element method can no longer meet the actual needs. Future 6G will have even higher requirements for computing power, and the problem of computing power bottlenecks in electromagnetic field simulation will become more prominent. Classical computing and algorithms are facing huge pressures. Compared with classical computers, quantum computers have advantages such as parallelism and exponential acceleration, and their computing power is extremely powerful. However, for simulating complex electromagnetic systems, traditional quantum algorithms such as QLA (Qubit lattice algorithm) cannot simulate Maxwell systems with source terms and various non-periodic physical boundary conditions, which makes quantum algorithms not practical. Summary of the Invention

[0003] In view of the above deficiencies in the prior art, the present invention proposes an electromagnetic simulation method in a homogeneous medium based on Schrödingerization, which can significantly reduce computing resources and solve the limitations of current quantum algorithm simulations.

[0004] The present invention relates to an electromagnetic simulation method based on Schrödingerization, including:

[0005] Step 1: According to the Maxwell equations in a homogeneous medium, obtain an equivalent matrix representation form;

[0006] Step 2: According to the classical space discretization format Yee format, process complex physical boundary conditions to obtain an ODE system;

[0007] Step 3: According to Schrödingerization, use the dimension-raising method to obtain a Hamiltonian system that can be calculated on a quantum computer;

[0008] Step 4: According to the Lie-Trotter-Suzuki method, construct a quantum circuit diagram for the Hamiltonian system obtained in Step 3 to record the process of applying a series of quantum operators to a quantum state in a graphical and compact manner.

[0009] Technical Effects

[0010] By transforming Gauss's law of the electrostatic field and the limiting conditions of magnetic flux continuity into time-evolution equations, a matrix representation of an 8×8 scale is obtained. Auxiliary variables are added to transform the inhomogeneous system after spatial discretization into a linear homogeneous ODE system. Using Schrödingerization, the original system is transformed into a Hamiltonian system that can be computed on a quantum computer. A quantum circuit diagram of the Hamiltonian system generated by Schrödingerization is constructed using the Bell basis and the Lie-Trotter-Suzuki method. The target variables are obtained by reasonably selecting the data on the auxiliary qubits for measurement. The quantum algorithm of the Hamiltonian system obtained by Schrödingerization in the present invention has the advantage of exponential acceleration compared with the traditional explicit FDTD algorithm. Restoring the original variables is simple and efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 is a flow chart of the present invention;

[0012] Figure 2 is a schematic diagram of a medium model of the electromagnetic field distribution;

[0013] Figure 3 is a quantum circuit diagram of Schrödingerization;

[0014] Figure 4 、 5 、6 is a schematic diagram of the implementation effect;

[0015] Figure 7 is a schematic diagram of the PML layer in Embodiment 2. DETAILED DESCRIPTION OF THE INVENTION

[0016] As Figure 1 shown, this embodiment relates to a simulation method of an electromagnetic field in a homogeneous medium with a source term current J(x) = [J Figure 2 shown, J x , J y , J z and a charge density ρ(x), including:

[0017] Step 1) Rewrite the original electromagnetic equation into a matrix form, and define new variables as:

[0018]

[0019] where: E = [E x , E y , E z is the electric field strength, B = [B x , B y , B x is the magnetic flux density, r 1 (t) ≡ r 2(t)≡0 are respectively the conservation error of Gauss's law and the residual of the magnetic flux divergence, ε and μ are respectively the permittivity and permeability of the medium,

[0020] is the speed of light in the medium, and the new variable F satisfies:

[0021]

[0022] Step 2) Use the finite difference format for spatial discretization and add an auxiliary variable b to obtain a homogeneous ordinary differential equation:

[0023] where: the discrete variable is The matrix A = A obtained by using the finite difference format for spatial discretization curl +A B , different spatial discretization methods and boundary conditions correspond to different matrices A B = 0, Different physical boundary conditions mean that: the matrix corresponding to the periodic boundary condition is:

[0024]

[0025] where: M is the number of grids in each direction, and the grid size is Δx = L x / M, Δy = L y / M, Δz = L z / M.

[0026] In this embodiment, the perfect electric conductor (PEC) boundary condition corresponds to A B = 0 and

[0027]

[0028] Step 3) Adopt Schrödingerization to transform the homogeneous ordinary differential equation into a convection equation, specifically including:

[0029] 3.1 Adopt Schrödingerization to first transform the original system into a convection equation:

[0030]

[0031] 3.2 Numerically discretize the p region, specifically including:

[0032] 3.2.1 Truncate the newly added extra region to a sufficiently large region [-L, R], where L and R are large enough to satisfy e -L ≈ 0, e -R ≈ 0, and where:

[0033] T is the simulation duration.

[0034] 3.2.2 Uniformly divide the p region into N parts to obtain the grid p j =-L + j(R + L) / N.

[0035] 3.2.3 Discretize in the p direction using the spectral method:

[0036] where: Φ is the Fourier transform matrix, which corresponds to the quantum Fourier transform in a quantum computer;

[0037] is the difference operator in the phase space; Here, g h is an N×1 vector g h =[g(p0)…g(p N-1 )] T .

[0038] Step 4) Construct the quantum circuit diagram of the Hamiltonian system after Schrödingerization on a quantum computer:

[0039] where: the auxiliary qubits selected by the quantum computer are measured multiple times, with probability e -2p* ||u b (T)|| 2 / u b (0)|| 2 to obtain the target variable.

[0040] As Figure 3 shown, the quantum Fourier transform and its inverse transform in the quantum circuit diagram are respectively where: the unitary transformation U = exp(-iHT).

[0041] Preferably, using the Lie-Trotter-Suzuki method, the unitary transformation is approximated as where: N t is the number of time steps.

[0042] After specific experiments, the PEC boundary condition is adopted. Considering the distribution of the electromagnetic field in a one-dimensional homogeneous medium, that is, the simplified version of the electromagnetic field equation is E=(0, E y (x,t), 0), B=(0, 0, B z (x,t)); the dielectric parameter and magnetic permeability of the medium are ε = μ = 1, the length of the medium L x =2, the calculation region is set as [0, 2], the two ends of the medium are perfect conductors, the initial electromagnetic field is a beam of pulses, and the calculation is set as The discretization size of the Yee grid is Δx = 2 / M = 2 -11 , the simulation duration T = 1.6, and the number of time steps is N t = 2 10 . In this embodiment, u b = [E h B h J h T . The corresponding spatial discrete matrix is

[0043] A B The truncation interval in the p direction is p ∈ [-15, 15], and the grid discretization size is Δp = 15 / 2 6 When restoring the target variable, the target variable u(T) is selected As Figure 4 shown, it can be seen that the simulation results obtained by Schrödingerization coincide with those of the classical algorithm.

[0044] After repeated experiments, the PML boundary condition is adopted. The size of the medium is L x = 10, the calculation region is [-5, 5], and the grid size is Δx = 5 / 2 8 . The initial electromagnetic field is set to 0, and the source term is a pulse function J = δ(x) fixed at the origin. During the calculation, the Gaussian function is used for approximation, that is PML layer σ h = [σ1,...σ M-1 has a length of 5, the grid size remains unchanged, and the medium diagram is as Figure 7 shown. The PML layer uses the dissipation coefficient σ = 5, and σ = 0 in the target region. The simulation end time T = 15, and the number of time grids is N t = 2 10 . In this embodiment, the corresponding discrete matrix is:

[0045] The truncation region in the p direction is [-75, 10], and the grid size is Δp = 85 / 2 10 . When restoring the original variable using the target variable u(T), the selected p * satisfies p * = min{p k > 8: p k = -75 + kΔp, 0 ≤ k ≤ 2 10}}. As Figure 5 , Figure 6 shown, it can be seen that the simulation results of the quantum algorithm after Schrödingerization coincide with those of the classical algorithm. In the PML layer, the electromagnetic field rapidly decays to zero.

[0046] ​In summary, under the same δ accuracy, for the existing classical explicit FDTD algorithm, the optimal computational complexity of a d-dimensional system (d = 1, 2, 3) is The present invention is only It can be seen that the present invention significantly improves the computational efficiency and reduces the computational time.

[0047] Those skilled in the art can make local adjustments to the above specific embodiments in different ways without departing from the principles and purposes of the present invention. The protection scope of the present invention is defined by the claims and is not limited by the above specific embodiments. All implementation solutions within its scope are subject to the present invention.

Claims

1. An electromagnetic simulation method in a homogeneous medium based on Schrödinger transformation, characterized in that: According to the Maxwell equations in homogeneous media, an equivalent matrix expression is obtained; according to the classical space discrete format Yee format, complex physical boundary conditions are processed to obtain an ODE system; according to Schrödinger transformation, the dimension-raising method is used to obtain a Hamiltonian system that can be calculated on a quantum computer; according to the Lie-Trotter-Suzuki method, a quantum circuit diagram is constructed for the Hamiltonian system to record the process of applying a series of quantum operators to quantum states in a graphical and compact way, including: Step 1) Rewrite the original electromagnetic equation into matrix form and define new variables as: Where: E = [E x ,E y ,E z ] is the electric field strength, B=[B x ,B y ,B z ] is the magnetic flux density, r 1 (t)≡r 2 (t)≡0 is the residual error of Gauss's law conservation and the magnetic flux divergence, ε and μ are the dielectric constant and magnetic permeability of the medium, respectively. is the speed of light in the medium, and the new variable satisfy: Step 2) Use the finite difference format for spatial discretization and add the auxiliary variable b to obtain the homogeneous ordinary differential equation: Among them: discrete variables are The matrix A obtained by spatial discretization using the finite difference format is A=A curl +A B , different spatial discretization methods and boundary conditions correspond to different matrices Different physical boundary conditions refer to: The matrix corresponding to the periodic boundary condition is: in: M is the number of grids in each direction, and the grid size is Δx = L x / M,Δy=L y / M,Δz=L z / M; Step 3) Using Schrödinger transformation, the homogeneous ordinary differential equation is first transformed into a convection equation, which specifically includes: 3.1 Using Schrödinger transformation, first transform the original system into a convection equation: 3.2 numerically discretize the p region; Step 4) Construct the quantum circuit diagram of the Schrodingerized Hamiltonian system on a quantum computer: Among them: Auxiliary bits selected by the quantum computer Multiple measurements, with probability Get the target variable.

2. The electromagnetic simulation method in a homogeneous medium based on Schrödinger transformation according to claim 1 is characterized in that Conductor PEC boundary condition corresponds to A B =0 and 3. The electromagnetic simulation method in a homogeneous medium based on Schrödinger transformation according to claim 1, characterized in that: The step 3.2 specifically includes: 3.2.1 Truncate the newly added extra area to a sufficiently large area [-L, R], where L and R are large enough to satisfy e -L ≈0,e -R ≈0, and in: T is the simulation duration; 3.2.2 Divide the p region uniformly into N parts and obtain the grid p j = -L + j (R + L) / N; 3.2.3 Discretization using spectral method in p direction: Where: Φ is the Fourier transform matrix, which corresponds to the quantum Fourier transform in the quantum computer; is the difference operator in phase space; Here h is an N×1 vector g h =[g(p0)…g(p N-1 )] T .

4. The electromagnetic simulation method in a homogeneous medium based on Schrödinger transformation according to claim 1, characterized in that: The quantum Fourier transform or inverse transform in the quantum circuit diagram is Where: unitary transformation U = exp(-iHT).

5. The electromagnetic simulation method in a homogeneous medium based on Schrödinger transformation according to claim 4 is characterized in that: Using the Lie-Trotter-Suzuki method, the unitary transformation is approximated by Where: N t is the number of time steps.

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