A quantum dynamics simulation method based on physical information subspace and riemann optimization

Abstract

This invention discloses a quantum dynamics simulation method based on physical information subspace and Riemann optimization, which solves the problems of insufficient expressive power, poor trainability, inefficient subspace construction, and insufficient utilization of physical constraints faced by existing technologies when simulating quantum dynamics on medium-scale quantum devices with noise. The method of this invention is carried out in two stages. In the first stage, a set of adjustable parameters is applied to m quantum circuits with identical structures and independent parameters to generate a set of ground states. These ground states span a subspace for approximating the real dynamics of the system, and the subspace has a low dimension, thereby compressing the complex quantum dynamic evolution into the subspace. In the second stage, a smooth manifold is set in the subspace, and Riemann gradient descent is used on the smooth manifold to minimize the residual of the time-dependent Schrödinger equation. It has a linear convergence rate and can avoid the barren plateau problem in the original high-dimensional parameter space, thereby achieving effective decoupling of expressive power and trainability.

Description

Technical Field

[0001] This invention relates to a cross-disciplinary approach combining quantum computing and machine learning, specifically a quantum dynamics simulation method based on physical information subspace and Riemann optimization. Background Technology

[0002] In fields such as quantum many-body physics and quantum chemistry, accurately simulating the dynamical evolution of quantum systems under time-dependent Hamiltonians is a core task. For noisy, medium-scale quantum devices (NISQ devices, with finite coherence times and limited circuit depth), existing technological approaches mainly fall into three categories:

[0003] The first category is the product formula method, such as Trotterization. This type of method approximates the evolution by decomposing the time evolution into local operations on short time steps. The concept is simple and easy to implement. However, the required circuit depth increases polynomially with the evolution time and accuracy, easily exceeding the coherence time limit on practical NISQ devices, making it difficult to use for long-term, high-precision simulations.

[0004] The second category is Variational Quantum Algorithms (VQA). These methods approximate time evolution using shallow parameterized quantum circuits (PQCs), updating parameters by solving equations of motion or globally minimizing a time loss function. However, VQA faces a dilemma of "expressive power versus trainability": highly expressive circuits are prone to barren plateaus, leading to vanishing gradients and difficulty in optimization; while shallow circuits, though easy to train, lack sufficient representational power to capture complex dynamics.

[0005] The third category is subspace projection methods, such as quantum Krylov methods and variable subspace methods. These methods confine the dynamics to a low-dimensional subspace to reduce circuit depth. However, existing subspaces are often constructed in ways independent of the problem (e.g., using power of the Hamiltonian, time evolution operators, or random states), failing to fully utilize the spectral structure of the target Hamiltonian or the causal dynamics from a known initial state. This results in the subspace dimension needing to grow rapidly to maintain accuracy, diminishing its resource advantage.

[0006] A common limitation of the above methods is that the optimization process is carried out in a flat parameter space with scarce physical information. Physical constraints (such as state normalization and Schrödinger equation residuals) are treated only as post-processing or penalty terms, rather than actively utilizing the intrinsic geometric structure of the quantum state space to guide the optimization. This lack of physical prior makes it difficult for the algorithms to simultaneously achieve expressive power and trainability with limited resources.

[0007] In the field of chemical materials, the above methods are difficult to efficiently simulate molecular dynamics, chemical reaction pathways, and non-equilibrium quantum processes in novel functional materials (such as high-temperature superconductors and topological insulators).

[0008] In the field of artificial intelligence, tasks such as quantum neural network training, quantum generative models, and quantum reinforcement learning also face problems such as low efficiency and poor scalability in dynamic simulation. Summary of the Invention

[0009] The purpose of this invention is to address the problems of insufficient expressive power, poor trainability, inefficient subspace construction, and insufficient utilization of physical constraints in existing technologies for simulating quantum dynamics on noisy medium-scale quantum devices, and to provide a quantum dynamics simulation method based on physical information subspaces and Riemann optimization.

[0010] To achieve the above objectives, the technical solution provided by this invention is as follows:

[0011] A quantum dynamics simulation method based on physical information subspace and Riemann optimization is characterized by the following steps:

[0012] Step 1: Select m quantum circuits with identical structures and independent parameters, where m ≥ 2. At time t, apply a set of adjustable parameters to the m quantum circuits that are initially in zero states to generate a set of ground states.

[0013] Step 2: Based on the set of ground states obtained in Step 1, design a loss function for the physical consistency constraint, diversity constraint and coverage constraint of the quantum circuit. Use the loss function to perform multi-objective pre-training on m quantum circuits to obtain a set of time-varying ground states with fixed adjustable parameters.

[0014] Step 3: In the time interval [0,T], represent the quantum state to be determined as a linear combination of time-varying ground states, and the quantum state to be determined satisfies the physical normalization condition at the target time T;

[0015] Step 4: Define a smooth manifold, and the combination coefficient vector of the linear combination lies on the smooth manifold. At the same time, design a physical information loss function for optimizing the combination coefficient vector on the smooth manifold.

[0016] Step 5: Initialize the combination coefficient vector and update it using the Riemann gradient descent method. Search for the combination coefficient vector that minimizes the physical information loss function as the optimal combination coefficient vector.

[0017] Step 6: Substitute the optimal combination coefficient vector into the linear combination to obtain the approximate quantum state at any time in the time interval [0,T], and complete the quantum dynamics simulation based on the physical information subspace and Riemann optimization.

[0018] Further, in step 1, the ground state is represented as:

[0019] ;

[0020] In the formula, This represents the i-th ground state at time t, where i = 1, 2, ..., m; This represents the adjustable parameter applied to the i-th ground state. Indicates adjustable parameters Evolution operators acting on quantum circuits This indicates that n qubits are in the ground state. The state of a composite system, n≥1.

[0021] Furthermore, the design loss function expression for the physical consistency constraint, diversity constraint, and coverage constraint of the quantum circuit described in step 2 is as follows:

[0022] ;

[0023] in, Let represent the average of the energy variances of the m ground states at the target time T. This represents the average value of the overlap modulus between different ground states. This represents the average of the squared magnitudes of the overlap between the m ground states and the known initial state at the initial time. , , These represent the weight coefficients of the physical consistency constraint, diversity constraint, and coverage constraint, respectively.

[0024] Furthermore, in step 2, the average value of the energy variance of the m ground states at the target time T. The expression is:

[0025] ;

[0026] In the formula, This represents the i-th ground state at time T. express Hermitian conjugate, This represents the Hamiltonian at the target time T;

[0027] Average value of overlapping modulus between different ground states The expression is:

[0028] ;

[0029] in, Indicates modulo, This represents the j-th ground state at time T.

[0030] The average of the squared magnitudes of the overlap between the m ground states and the known initial states at the initial time. The expression is:

[0031] ;

[0032] in, This represents the Hermitian conjugate of the i-th ground state at the initial moment. This represents the initial state of a known quantum state to be determined.

[0033] Furthermore, step 3 specifically involves:

[0034] The quantum state to be determined is expressed as a linear combination of time-varying ground states, as follows:

[0035] ;

[0036] In the formula, Let c represent the quantum state to be determined at time t, and let c represent the combination coefficient vector, with values ​​of 100,000 and 200,000 respectively. transpose, Represents the i-th combination coefficient;

[0037] The quantum state to be determined at the target time T is represented as: The quantum state to be determined satisfies the physical normalization condition at the target time T, expressed as:

[0038] ;

[0039] This then transforms into:

[0040] ;

[0041] in, It is the transpose of c. This represents the overlap matrix formed by the inner products of the ground states.

[0042] Furthermore, in step 3, the first term in the overlapping matrix formed by the ground state inner product... i Line 1 j Column elements The expression is:

[0043] ;

[0044] In the formula, i The range of values ​​for is 1, ..., m ; j The range of values ​​for is 1, ..., m .

[0045] Furthermore, step 4 specifically involves:

[0046] Define a smooth manifold And the combination coefficient vector of the linear combination lies on a smooth manifold. Above, represented as:

[0047] ;

[0048] In the formula, Represents complex Euclidean space;

[0049] Design a physical information loss function for optimizing the combined coefficient vector on smooth manifolds. , is represented as:

[0050] ;

[0051] In the formula, Represents the regularization parameter. This represents the total number of discrete time points selected on [0,T]. Indicates the first A discrete time point, express The Hamiltonian at time t. express The quantum state is always waiting to be determined. Represents the quantum state to be determined at the initial time. i represents an imaginary number.

[0052] Furthermore, step 5 specifically includes:

[0053] Step 5.1: Preset the norm threshold and maximum number of iterations for the Riemann gradient, and initialize the combined coefficient vector;

[0054] Step 5.2: Calculate the Euclidean gradient of the physical information loss function with respect to the current combined coefficient vector c at discrete time points. ;

[0055] Step 5.3: Convert the Euclidean gradient. Projecting onto the tangent space of the smooth manifold at the current discrete time point yields the Riemann gradient. ;

[0056] Step 5.4: Move along the negative Riemann gradient direction with a step size of Update the combination coefficient vector c to obtain a temporary vector. The expression is:

[0057] ;

[0058] Step 5.5: For the temporary vector Perform a shrinkage operation and map it back onto the smooth manifold as a new combination coefficient vector c;

[0059] Step 5.6: Determine whether the norm of the Riemann gradient at the current discrete time point is less than the norm threshold of the Riemann gradient or whether the maximum number of iterations has been reached;

[0060] If so, then the optimal combination coefficient vector is the combination coefficient vector c when the norm of the Riemann gradient is minimized.

[0061] If not, update the discrete time points and return to step 5.2.

[0062] Furthermore, in step 5.3, the Riemann gradient The expression is:

[0063] ;

[0064] This indicates taking the real part.

[0065] Further, in step 5.5, the temporary vector... To perform a shrinkage operation, the expression is:

[0066] ;

[0067] In the formula, Represents the temporary vector The result obtained by performing the retraction operation.

[0068] Compared with the prior art, the present invention has the following beneficial technical effects:

[0069] 1. This invention provides a quantum dynamics simulation method based on physical information subspace and Riemann optimization. It is a two-stage physical information quantum dynamics simulation method. In the first stage, a set of adjustable parameters are applied to m quantum circuits with identical structures and independent parameters to generate a set of ground states. These ground states span a subspace used to approximate the real dynamics of the system, and the subspace has a low dimension. This compresses the complex quantum dynamic evolution into the subspace. In the second stage, based on the first stage, Riemann optimization is performed in the subspace, which can avoid the barren plateau problem in the original high-dimensional parameter space, thereby achieving effective decoupling of expressive power and trainability when simulating quantum dynamics on medium-scale quantum devices.

[0070] 2. The present invention provides a quantum dynamics simulation method based on physical information subspace and Riemann optimization, which performs Riemann gradient descent on a smooth manifold to minimize the residual of the time-dependent Schrödinger equation, has a linear convergence rate, and can improve simulation efficiency.

[0071] 3. The quantum dynamics simulation method based on physical information subspace and Riemann optimization provided by this invention has been verified (numerical experiments on the transverse field Ising model) to maintain a fidelity of over 99% throughout the entire evolution process, and the energy error is two orders of magnitude lower than that of traditional variational quantum algorithms. Furthermore, this method maintains low error even when the system crosses the quantum critical point, demonstrating good robustness to quantum phase transitions. Attached Figure Description

[0072] Figure 1 The figure shows a comparison of the evolution curves of the string order parameters over time in the actual state (Exact) calculated using the quantum dynamics simulation method (PIQS) based on physical information subspace and Riemann optimization and the exact diagonalization method of this invention. Figure 1 In the diagram, (a) represents a NISQ device with size A=7; (b) represents a NISQ device with size A=8; (c) represents a NISQ device with size A=9; and (d) represents a NISQ device with size A=10.

[0073] Figure 2 The graph shows a comparison of the absolute error of the reference value over time using the Physical Information Subspace and Riemann Optimization Quantum Dynamics Simulation Method (PIQS) of this invention, the TDVQA method, and the MPS-TDVP method. Figure 2 In the diagram, (a) represents a NISQ device with size A=7; (b) represents a NISQ device with size A=8; (c) represents a NISQ device with size A=9; and (d) represents a NISQ device with size A=10.

[0074] Figure 3 The graph shows a comparison of the fidelity evolution over time of the quantum dynamics simulation method (PIQS) based on physical information subspace and Riemann optimization, as well as the TDVQA and MPS-TDVP methods. Figure 3 In the diagram, (a) represents a NISQ device with size A=7; (b) represents a NISQ device with size A=8; (c) represents a NISQ device with size A=9; and (d) represents a NISQ device with size A=10.

[0075] Figure 4 The graph shows a comparison of the energy relative error evolution over time between the Quantum Dynamics Simulation Method (PIQS) based on Physical Information Subspace and Riemann Optimization (as described in this invention) and the TDVQA and MPS-TDVP methods. Figure 4 In the diagram, (a) represents the NISQ device size A=7; (b) represents the NISQ device size A=8; (c) represents the NISQ device size A=9; and (d) represents the NISQ device size A=10. Detailed Implementation

[0076] To make the objectives, advantages, and features of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. Those skilled in the art should understand that these embodiments are merely used to explain the technical principles of the present invention and are not intended to limit the scope of protection of the present invention.

[0077] This embodiment provides a quantum dynamics simulation method based on physical information subspace and Riemann optimization, including the following steps:

[0078] Step 1: Select m quantum circuits with identical structures and independent parameters, where m ≥ 2. At time t, apply a set of adjustable parameters to the m quantum circuits in their initial zero states to generate a set of ground states. These ground states will span a subspace used to approximate the true dynamics of the system, and this subspace has a low dimension.

[0079] The ground state is represented as:

[0080] ;

[0081] In the formula, This represents the i-th ground state at time t, where i = 1, 2, ..., m; This represents the adjustable parameter applied to the i-th ground state. Indicates adjustable parameters Evolution operators acting on quantum circuits This indicates that n qubits are in the ground state. The state of a composite system, n≥1.

[0082] Step 2: To obtain a set of high-quality ground states, this embodiment performs multi-objective pre-training on the above m quantum circuits to optimize their tunable parameters, so that the m quantum circuits simultaneously satisfy the following constraints:

[0083] (a) Physical consistency constraints

[0084] The energy variance of each ground state at the target time T is calculated and expressed as: ;

[0085] in, This represents the i-th ground state at time T. express Hermitian conjugate, This represents the Hamiltonian at the target time T.

[0086] The average of the energy variances of m ground states at the target time T The expression is:

[0087] .

[0088] In the multi-objective pre-training process, the goal is to minimize the average energy variance of the m ground states at the target time T. The goal is to ensure that each ground state approximates the instantaneous eigenstate of the Hamiltonian H(T), thereby providing a physical basis for expressing the low-energy dynamics of the system.

[0089] (ii) Diversity constraints

[0090] The formula for calculating the square of the overlap modulus between different ground states is: ;

[0091] in, Indicates modulo, This represents the j-th ground state at time T.

[0092] Average value of overlapping modulus between different ground states The expression is:

[0093] .

[0094] For all The goal is to minimize the average value of the overlap modulus. The diversity constraint encourages the ground states to be as linearly independent as possible, preventing them from coinciding in the subspace, thus ensuring the overlap matrix formed by the inner products of the ground states. (its elements) It has a good condition number, which provides numerical stability for subsequent optimization.

[0095] (iii) Coverage constraints

[0096] Calculate the difference between each ground state at the initial time (t=0) and the known initial state. The goal is to maximize the square of the overlap modulus of m ground states at the initial time and the known initial state. The expression is:

[0097] ;

[0098] in, This represents the Hermitian conjugate of the ground state at the initial moment. This represents the initial state of a known quantum state to be determined.

[0099] Based on the above three constraints, the loss function expression is designed as follows:

[0100] ;

[0101] in, , , These represent the weight coefficients of the physical consistency constraint, diversity constraint, and coverage constraint, respectively. Multi-objective pre-training is performed on m quantum circuits using a loss function, searching for the adjustable parameters corresponding to the minimum loss function, resulting in a set of time-varying ground states with fixed adjustable parameters.

[0102] Step 3: In the time interval [0, T], represent the quantum state to be determined as a linear combination of time-varying ground states, and the quantum state to be determined satisfies the physical normalization condition at the target time T; specifically:

[0103] The quantum state to be determined is expressed as a linear combination of time-varying ground states, as follows:

[0104] ;

[0105] In the formula, Let c represent the quantum state to be determined at time t, and let c represent the combination coefficient vector, with values ​​of 100,000 and 200,000 respectively. transpose, Represents the i-th combination coefficient;

[0106] The quantum state to be determined at the target time T is represented as: The quantum state to be determined satisfies the physical normalization condition at the target time T, expressed as:

[0107] ;

[0108] This then transforms into:

[0109] ;

[0110] in, It is the transpose of c. Let represent the overlap matrix formed by the inner products of the ground states, where the _i_th ... i Line 1 j Column elements The expression is:

[0111] ;

[0112] In the formula, i The range of values ​​for is 1, ..., m ; j The range of values ​​for is 1, ..., m .

[0113] Step 4: Define a smooth manifold And the combination coefficient vector of the linear combination lies on a smooth manifold. Above, represented as:

[0114] ;

[0115] In the formula, Let this be a complex Euclidean space, where all physically allowed combination coefficient vectors lie on this smooth manifold. superior.

[0116] To evaluate the physical rationality of candidate combination coefficient vectors, a physical information loss function is designed for optimizing combination coefficient vectors on smooth manifolds. , is represented as:

[0117] ;

[0118] In the formula, Represents the regularization parameter ( >0), Indicates a known initial state. This represents the total number of discrete time points selected on [0,T]. Indicates the first A discrete time point, express The Hamiltonian at time t. express The quantum state is always waiting to be determined. This represents the quantum state to be determined at the initial moment. i represents an imaginary number.

[0119] The above physical information loss function It contains two terms: the first term is the mean square norm of the residuals of the time-dependent Schrödinger equation, and the second term is the initial state constraint term.

[0120] Step 5: Initialize the combination coefficient vector and update it using the Riemann gradient descent method to find the optimal combination coefficient vector that minimizes the physical information loss function; specifically:

[0121] Step 5.1: Initialize the combined coefficient vector on the smooth manifold by setting the norm threshold and maximum number of iterations for the Riemann gradient.

[0122] Step 5.2: Calculate the Euclidean gradient of the physical information loss function with respect to the current combined coefficient vector c at discrete time points. Euclidean gradient It can be obtained by combining the parameter shift rule on the quantum circuit with measurement.

[0123] Step 5.3: Convert the Euclidean gradient. Projecting onto the tangent space of the smooth manifold at the current discrete time point yields the Riemann gradient. The projection expression is:

[0124] ;

[0125] This indicates taking the real part.

[0126] This projection ensures that the search direction is aligned with the geometry of the smooth manifold, thereby automatically maintaining the normalization constraints in subsequent updates.

[0127] Step 5.4: Move along the negative Riemann gradient direction with a step size of Update the combination coefficient vector c to obtain a temporary vector. The expression is:

[0128] .

[0129] Step 5.5, due to the temporary vector Generally no longer satisfied It is necessary to process the temporary vector. Perform a shrinkage operation and map the new combination coefficient vector c back onto the smooth manifold; the expression is:

[0130] ;

[0131] In the formula, Represents the temporary vector The result of the shrinking operation is used as a new combination coefficient vector c and mapped back onto the smooth manifold.

[0132] This shrinkage operation ensures that the new combination coefficient vector lies strictly on the smooth manifold. Above, that is, the normalization constraint is satisfied.

[0133] Step 5.6: Determine if the condition is met.

[0134] Is the norm of the Riemann gradient at the current discrete time point less than the norm threshold of the Riemann gradient or has reached the maximum number of iterations?

[0135] If so, then the optimal combination coefficient vector c is the one whose norm of the Riemann gradient is minimized. ;

[0136] If not, update the discrete time points and return to step 5.2.

[0137] Step 6: Convert the optimal combination coefficient vector By substituting linear combinations, approximate quantum states at any time interval [0,T] are obtained, completing the entire process from quantum data acquisition (ground state preparation and measurement) to classical optimization (Riemann gradient descent), thus realizing high-fidelity and scalable quantum dynamics simulation.

[0138] This embodiment uses the Transverse-Field Ising Model (TFIM) as an example to illustrate in detail the application of the method of the present invention in simulating quantum quenching dynamics.

[0139] I. System and Task Settings

[0140] Consider a one-dimensional transverse-field Ising chain, whose Hamiltonian is expressed as follows: the first term is the interaction between adjacent spins in the z-direction, with strength J; the second term is the transverse field term, with strength g. In this embodiment, J is set to 1, and g is initially set to 2.0. At time t = 0, the transverse field is instantaneously changed (quenched) until g finally equals 0.5. The total evolution time T is 2.0, and the system contains 10 qubits. The initial state is taken as the transverse-field ground state, which can be prepared using a simple quantum circuit. The goal of this embodiment is to simulate, with high fidelity, the time-dependent quantum state driven by the quenched Hamiltonian, starting from the initial state.

[0141] II. Subspace Construction

[0142] Choosing m=8, each ground state is generated by a hardware-efficient parameterized quantum circuit with 3 layers, and the parameters of each circuit are randomly initialized.

[0143] Eight quantum circuits were pre-trained using a multi-objective method. A loss function for this multi-objective pre-training was designed, comprising three terms: physical consistency constraint, diversity constraint, and coverage constraint. The specific expression is as follows:

[0144] ;

[0145] in, , , .

[0146] The gradient of the loss function with respect to each quantum circuit parameter is calculated using the parameter-shift rule. The Adam optimizer is used for iterative optimization, and a total of 500 training rounds are performed until the loss function converges. The adjustable parameters applied to each quantum circuit are determined, resulting in a set of time-varying ground states with fixed adjustable parameters.

[0147] Uniformly discretized over the time interval [0, T] At discrete time points, the combination coefficient vector on the smooth manifold is initialized. The initial value of the combination coefficient vector c is determined by solving a least squares problem, so that the approximate state at the initial time is as close as possible to the true initial state. Then, the combination coefficient vector is updated using the Riemann gradient descent method. The combination coefficient vector that minimizes the physical information loss function is searched as the optimal combination coefficient vector. The optimal combination coefficient vector is substituted into the linear combination to obtain the approximate quantum state at any time in the time interval [0,T]. This completes the quantum dynamics simulation based on the physical information subspace and Riemann optimization.

[0148] III. Result Evaluation

[0149] Using the exact diagonalization method to calculate the true state (Exact) as a benchmark, the evolution curves of the chord sequence parameter over time under different NISQ device sizes are compared with the method of this embodiment (PIQS) as shown in the figure. Figure 1 As shown; where (a)-(d) correspond to NISQ device dimensions A=7, 8, 9, 10 respectively; according to Figure 1 It can be seen that the sine sequence parameter evolution curves of this embodiment (PIQS) and the actual state (Exact) under different NISQ device sizes coincide, indicating that the simulation accuracy of this embodiment (PIQS) is high.

[0150] The absolute error of the corresponding chord sequence parameter relative to the reference value obtained by the method (PIQS), TDVQA method, and MPS-TDVP method in this embodiment over time under different NISQ device sizes is shown in the following curves. Figure 2 As shown; where (a)-(d) correspond to NISQ device dimensions A=7, 8, 9, 10 respectively; according to Figure 2 As can be seen from (a)-(d), the absolute error of the chord sequence parameter of the method (PIQS) in this embodiment relative to the reference value is significantly lower than that of the TDVQA method and the MPS-TDVP method relative to the reference value.

[0151] The fidelity evolution curves obtained by the method (PIQS), TDVQA method, and MPS-TDVP method in this embodiment under different NISQ device sizes are shown below. Figure 3 As shown; where (a)-(d) correspond to NISQ device dimensions A=7, 8, 9, 10 respectively; according to Figure 3 As can be seen from (a)-(d), the fidelity of the method (PIQS) in this embodiment is always higher than 0.99, and the fidelity at the final moment is better than the results of using the TDVQA method and the MPS-TDVP method.

[0152] The energy relative error evolution curves obtained by the method (PIQS), TDVQA method, and MPS-TDVP method in this embodiment under different NISQ device sizes are shown below. Figure 4 As shown, according to Figure 4 It can be seen that the energy relative error of the method in this embodiment (PIQS) is about two orders of magnitude lower than that of the TDVQA method and the MPS-TDVP method.

[0153] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the present invention.

Claims

1. A quantum dynamics simulation method based on physical information subspace and Riemann optimization, characterized in that, Includes the following steps: Step 1: Select m quantum circuits with identical structures and independent parameters, where m ≥ 2. At time t, apply a set of adjustable parameters to the m quantum circuits that are initially in zero states to generate a set of ground states. Step 2: Based on the set of ground states obtained in Step 1, design a loss function for the physical consistency constraint, diversity constraint and coverage constraint of the quantum circuit. Use the loss function to perform multi-objective pre-training on m quantum circuits to obtain a set of time-varying ground states with fixed adjustable parameters. Step 3: In the time interval [0,T], represent the quantum state to be determined as a linear combination of time-varying ground states, and the quantum state to be determined satisfies the physical normalization condition at the target time T; Step 4: Define a smooth manifold, and the combination coefficient vector of the linear combination lies on the smooth manifold. At the same time, design a physical information loss function for optimizing the combination coefficient vector on the smooth manifold. Step 5: Initialize the combination coefficient vector and update it using the Riemann gradient descent method. Search for the combination coefficient vector that minimizes the physical information loss function as the optimal combination coefficient vector. Step 6: Substitute the optimal combination coefficient vector into the linear combination to obtain the approximate quantum state at any time in the time interval [0,T], and complete the quantum dynamics simulation based on the physical information subspace and Riemann optimization.

2. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 1, characterized in that: In step 1, the ground state is represented as: ； In the formula, This represents the i-th ground state at time t, where i = 1, 2, ..., m; This represents the adjustable parameter applied to the i-th ground state. Indicates adjustable parameters Evolution operators acting on quantum circuits This indicates that n qubits are in the ground state. The state of a composite system, n≥1.

3. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 2, characterized in that, The design loss function expression for the physical consistency constraint, diversity constraint, and coverage constraint of quantum circuits described in step 2 is as follows: ； in, Let represent the average of the energy variances of the m ground states at the target time T. This represents the average value of the overlap modulus between different ground states. This represents the average of the squared magnitudes of the overlap between the m ground states and the known initial state at the initial time. , , These represent the weight coefficients of the physical consistency constraint, diversity constraint, and coverage constraint, respectively.

4. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 3, characterized in that, In step 2, the average value of the energy variance of the m ground states at the target time T The expression is: ； In the formula, This represents the i-th ground state at time T. express Hermitian conjugate, This represents the Hamiltonian at the target time T; Average value of overlapping modulus between different ground states The expression is: ； in, Indicates modulo, This represents the j-th ground state at time T. The average of the squared magnitudes of the overlap between the m ground states and the known initial states at the initial time. The expression is: ； in, This represents the Hermitian conjugate of the i-th ground state at the initial moment. This represents the initial state of a known quantum state to be determined.

5. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 4, characterized in that, Step 3 specifically involves: The quantum state to be determined is expressed as a linear combination of time-varying ground states, as follows: ； In the formula, Let c represent the quantum state to be determined at time t, and let c represent the combination coefficient vector, with values ​​of 100,000 and 200,000 respectively. transpose, Represents the i-th combination coefficient; The quantum state to be determined at the target time T is represented as: The quantum state to be determined satisfies the physical normalization condition at the target time T, expressed as: ； This then transforms into: ； in, It is the transpose of c. This represents the overlap matrix formed by the inner products of the ground states.

6. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 5, characterized in that, In step 3, the overlapping matrix formed by the ground state inner product is the first... i Line 1 j Column elements The expression is: ； In the formula, i The range of values ​​for is 1, ..., m ; j The range of values ​​for is 1, ..., m .

7. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 5, characterized in that, Step 4 specifically involves: Define a smooth manifold And the combination coefficient vector of the linear combination lies on a smooth manifold. Above, represented as: ； In the formula, Represents complex Euclidean space; Design a physical information loss function for optimizing the combined coefficient vector on smooth manifolds. , represented as: ; In the formula, Represents the regularization parameter. This represents the total number of discrete time points selected on [0,T]. Indicates the first A discrete time point, express The Hamiltonian at time t. express The quantum state is always waiting to be determined. Represents the quantum state to be determined at the initial time. i represents an imaginary number.

8. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 7, characterized in that, Step 5 specifically involves: Step 5.1: Preset the norm threshold and maximum number of iterations for the Riemann gradient, and initialize the combined coefficient vector; Step 5.2: Calculate the Euclidean gradient of the physical information loss function with respect to the current combined coefficient vector c at discrete time points. ; Step 5.3: Convert the Euclidean gradient. Projecting onto the tangent space of the smooth manifold at the current discrete time point yields the Riemann gradient. ; Step 5.4: Move along the negative Riemann gradient direction with a step size of Update the combination coefficient vector c to obtain a temporary vector. The expression is: ; Step 5.5: For the temporary vector Perform a shrinkage operation and map it back onto the smooth manifold as a new combination coefficient vector c; Step 5.6: Determine whether the norm of the Riemann gradient at the current discrete time point is less than the norm threshold of the Riemann gradient or whether the maximum number of iterations has been reached; If so, then the optimal combination coefficient vector is the combination coefficient vector c when the norm of the Riemann gradient is minimized. If not, update the discrete time points and return to step 5.

2.

9. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 8, characterized in that: In step 5.3, the Riemann gradient The expression is: ; This indicates taking the real part.

10. The quantum dynamics simulation method based on physical information subspace and Riemann optimization according to claim 8, characterized in that: In step 5.5, the temporary vector To perform a shrinkage operation, the expression is: ; In the formula, Represents the temporary vector The result obtained by performing the retraction operation.

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