Fast method for solving eigen-spectrum of hamiltonian based on limited energy-topological charge constraint

Abstract

The application discloses a kind of based on limited energy-topological charge constraint Hamiltonian eigen-spectrum fast solution method and system, belong to quantum simulation and computational physics technical field.The application will topological charge zero condition as physical constraint term superimposed to Hamiltonian, by iteration solving eigen-spectrum of equivalent Hamiltonian, automatically exclude non-physical eigenstate in solving process, accelerate convergence to meet the physical eigen-spectrum of limited energy boundary condition.The application is applicable to quantum many-body system, quantum chemistry electronic structure calculation, condensed state physics energy band analysis etc.Situ, can improve convergence speed.

Description

Technical Field

[0001] This invention belongs to the field of quantum simulation and computational physics, and is particularly applicable to the calculation of eigenspectral density of quantum many-body systems, electronic structure calculation in quantum chemistry, band structure analysis in condensed matter physics, and Hamiltonian simulation in quantum computing. Specifically, it relates to a system and method for accelerating the solution of Hamiltonian eigenspectral density using the zero-topological-charge condition of prime number spectrum manifold. Background Technology

[0002] Existing methods for solving Hamiltonian eigenspectroscopy (such as the Lanczos algorithm, Davidson method, density matrix renormalization group, and exact diagonalization) have the following inherent limitations: Matrix size limitation: The memory footprint and computation time of the exact diagonalization method vary with the matrix dimension. Present to Growth, the actual maximum achievable dimension is approximately Much smaller than that required by quantum many-body systems Magnitude. Iterative methods lack physical constraints: Lanczos and Davidson methods approximate eigenvalues ​​through Krylov subspaces or subspace projections, but the iterative process relies solely on matrix-vector multiplication and lacks global constraints from the physical context (such as finite energy and topological charge conservation) to accelerate convergence or filter out non-physical eigenvalues. Existing spectral analysis does not utilize the distribution law of Riemann zeros: there is a correspondence between the zero spectrum of the Riemann zeta function and the energy level statistics of the quantum chaotic system, but existing methods for solving the eigenspectrum do not use this correspondence as a priori constraint. The distinction between ground state and excited state depends on post-processing: traditional methods distinguish between ground state and excited state by sorting eigenvalues, lacking the ability to actively select physical states during the solution process. This invention proposes a method for solving the Hamiltonian eigenspectrum based on finite energy-topological charge constraints. It uses the condition that the total topological charge on the prime spectral manifold returns to zero as a physical filter to automatically eliminate non-physical eigensoles during the iteration process, thereby accelerating convergence to the eigenspectrum that satisfies the finite energy boundary conditions. Summary of the Invention

[0003] This addresses the technical problems of existing Hamiltonian eigenspectral solving methods, such as limited matrix size, lack of physical constraint guidance, and slow iterative convergence. Technical solution This invention is based on the following mathematical fact: the ground state and low-excitement state of a physically realizable quantum system must satisfy the condition that the total topological charge returns to zero. (Equivalent to finite energy boundary conditions). The Hamiltonian... eigenstates Mapped to a prime spectral manifold Calculate the total topological load corresponding to the projected cross section. Only when satisfied Only the eigenstates of the eigenstates are preserved as physically permissible states. This system consists of four main components: Eigenstate-Topological Charge Mapping Module: Configured to map Hamiltonians Each candidate eigenstate Mapped to a prime spectral manifold Calculate the total topological load of the vortex distribution on it. The specific mapping method is as follows: the phase information of the eigenstates is mapped... Encoded as phase angle on the torus Scale coordinates . Energy-Topology Constraint Joint Optimization Module: Configured to simultaneously optimize during the energy minimization process. and constraint terms Construct an equivalent Hamiltonian ,in For topological load operators, This is the constraint strength coefficient. Iterative eigenspectrum solving module: Configured to solve using iterative methods (Lanczos, Davidson, or Jacobi-Davidson). The eigenspectrum is automatically excluded in each iteration. The eigensols retain the condition that satisfy... The physical state of. Convergence Determination and Output Module: Configured to determine whether the eigenvalue sequence converges; if converged, output that it satisfies the condition. If the intrinsic spectrum (including ground state energy and excited state energy) does not converge, the iteration continues. Overall process: Input Hamiltonian → Constructing an equivalent Hamiltonian → Iterative solution of eigenspectrum → Filtering State → Output physical eigenspectrum Physical correctness guarantees: - The condition for the total topological charge to return to zero is guaranteed by the Chern-Gauss-Bonnet theorem, excluding non-physical states; - eigenvalues Eigenvalues ​​of the original Hamiltonian The relationship between them is ,when When large enough, non-zero The state is pushed to the high-energy region without affecting the low-energy physics spectrum; Selection principle: greater than the system bandgap Double the amount to ensure effective screening. Beneficial effects The effective eigenspectral space is compressed by filtering out non-physical states through topological constraints. The above accelerates convergence; The distinction between the ground state and the excited state is completed automatically during the solution process, without the need for post-processing sorting; Applicable to any Hermitian Hamiltonian (quantum many-body, quantum chemistry, condensed matter physics) without modifying the underlying algorithm framework; It shares the same set of topology load computing hardware acceleration units as RH00207; The intrinsic spectrum output automatically satisfies the finite energy boundary conditions and can be directly used for subsequent physical analysis. Terminology Explanation

[0004] Terminology Meaning Total topological load zeroing condition The global constraint that the physically acceptable state of the system must satisfy is equivalent to the finite energy boundary condition, and is guaranteed by the Chern-Gauss-Bonnet theorem. Equivalent Hamiltonian : Add topological load constraint terms to the original Hamiltonian, so that the iterative solution process automatically selects those that satisfy the constraints. The eigenstates. Topological charge operator The operator that maps eigenstates to vortex distributions and computes the total topological load has eigenvalues. It is an integer. Attached Figure Description

[0005] Figure 1 This is a flowchart illustrating the overall process of this invention. Figure 2 Comparison of eigenspectral density with and without topological constraints; Figure 3 Constraint strength coefficient The curve showing the effect of convergence speed; Figure 4 This is a comparison chart of convergence speeds for different system sizes. Detailed Implementation

[0006] Example 1: Calculation of Ground State Energy of Quantum Many-Body System (Heisenberg Chain) This embodiment is applicable to the accurate calculation of the ground state energy of a one-dimensional antiferromagnetic Heisenberg chain. Step S1, Hamiltonian Input: Input length spin Antiferromagnetic Heisenberg chain Hamiltonian ( ). Step S2, Topological Charge Mapping Rule Configuration: Set mapping rules: Map each spin state The phase information is mapped to toroidal coordinates. The vortex distribution is statistically analyzed when traversing all spin configurations. Step S3, Constructing the Equivalent Hamiltonian: ,in . Step S4, Iterative Solution: Solve using the Lanczos algorithm. The minimum eigenvalue (ground state energy) is calculated, and the total topological charge of the candidate eigenstates is calculated in each iteration. Only retain The state. Step S5, Convergence and Output: When the energy change is less than Stop iteration when the time is right and output the ground state energy. and the ground state eigenvectors. Technical effect: For For systems like this, the traditional Lanczos algorithm requires... The invention requires convergence in the next iteration. Next, accelerate approximately And the output ground state automatically satisfies . Example 2: Quantum Chemical Electronic Structure Calculation (Water Molecule) This embodiment is applicable to electronic structure calculations for small molecule systems. Step C1, Hamiltonian Input: Input the electronic structure Hamiltonian of the water molecule (using the STO-3G basis set, approximately...). (dimensional matrix). Step C2, Topological constraint configuration: Map each molecular orbital configuration to a vortex distribution on a prime spectral manifold. Step C3, Eigenspectrum Determination: Solve using the Davidson method. The lowest Each eigenvalue. Step C4, Physical state output: The output satisfies The eigenvalue spectrum excludes non-physical excited states. Technical effect: In On a 3D matrix, the computation time is reduced from approximately The price dropped to approximately [number] seconds. The time is seconds, and the output excited state energy level is physically reasonable (there are no non-physical negative states). (state interference). Example 3: Topological band structure analysis (in conjunction with RH00207) This embodiment is applicable to the band structure calculation of topological insulators / semi-metals. Step T1, Hamiltonian Input: Input the valid value of the topological insulator Bi2Se3. Hamiltonian. Step T2, Constraint Solving: In Automatically apply during spatial scanning constraint. Step T3, Band Output: Output the topological band structure that satisfies the finite energy boundary conditions. Technical effect: While ensuring correct band topology classification, the computational speed is improved by approximately [percentage missing]. It can also automatically eliminate non-physical stray energy bands.

Claims

1. A fast solution method for the eigenspectrum of Hamiltonian based on finite energy-topological charge constraints, characterized in that, Includes the following steps: Equivalent Hamiltonian construction steps: The processor converts the target Hamiltonian... With topological load constraints Superposition to construct an equivalent Hamiltonian ,in For topological load operators, The constraint strength coefficients are used; the iterative eigenspectra solution steps are as follows: an iterative method is employed to solve... The eigenspectrum is used to calculate the total topological load of the candidate eigenstates in each iteration. Only retain those that meet the requirements. The physically permissible states, in which Preset tolerance; Output steps: Output the eigenspectrum that satisfies the constraints.

2. The method according to claim 1, characterized in that, The topological load operator The construction method is as follows: candidate eigenstates Mapped to a prime spectral manifold vortex distribution on the topology, calculate the total topological load. ,in For local vortex topological loads.

3. The method according to claim 1, characterized in that, The constraint strength coefficient The value is greater than the target Hamiltonian energy gap. times.

4. The method according to claim 1, characterized in that, The preset tolerance The range of values ​​is to .

5. The method according to claim 1, characterized in that, The iterative method is any one of the Lanczos algorithm, the Davidson method, the Jacobi-Davidson method, or the Arnoldi method.

6. The method according to claim 1, characterized in that, The intrinsic spectrum includes ground state energy and excited state energy, satisfying... The state is labeled as a physical state, non-zero. The state is marked as a non-physical state and its energy offset is recorded.

7. A fast solution system for Hamiltonian eigenspectroscopy based on finite energy-topological charge constraints, characterized in that, include: Hamiltonian input module, configured to receive target Hamiltonian ; The topological charge operator constructor module is configured to construct operators that map eigenstates to topological charges. ; The equivalent Hamiltonian construction module is configured to build... ; The iterative solution module is configured to solve... The eigenspectrum is used to screen physical states; The output module is configured to output the eigenspectrum that satisfies the constraints.

8. The system according to claim 7, characterized in that, The topological charge operator construction module is implemented in the form of a hardware accelerator and shares hardware units with the curvature global integral module of RH00200.

9. The system according to claim 7, characterized in that, The system is deployed in a quantum chemistry computing software package, a condensed matter physics simulation platform, or a quantum computing simulation framework.

10. The system according to claim 7, characterized in that, The eigenspectrum output by the system can be directly used for subsequent physical analysis without additional filtering or post-processing.

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