A method for analyzing physical properties of a strong coupling system and related devices

Abstract

The application discloses a physical property analysis method of a strongly correlated system and related equipment, relates to the technical field of physical property analysis based on quantum computing, and comprises the following steps: obtaining a parameterized variational quantum circuit according to the Hamiltonian of a target strongly correlated system and optimizing the parameterized variational quantum circuit; embedding the optimized variational quantum circuit as a time evolution submodule into a preset Green function measurement circuit, optimizing the parameters of a diagonal part in the optimized variational quantum circuit through scaling, constructing a target quantum circuit and running the target quantum circuit, performing frequency domain transformation on time domain data of the Green function, and calculating the state density of the strongly correlated system; and analyzing the physical property of the strongly correlated system based on the state density. The application precisely extends to the calculation of the Green function and the state density, which are core physical quantities in the analysis of the strongly correlated system, solves the problem that the existing variational algorithm cannot efficiently obtain a dynamic correlation function, and provides a new path for analyzing the physical property of the strongly correlated system.

Description

Technical Field

[0001] This invention relates to the field of physical property analysis technology based on quantum computing, and in particular to a method and related equipment for analyzing the physical properties of strongly correlated systems. Background Technology

[0002] The physical properties of strongly correlated systems (such as high-temperature superconductors and Mott insulators), including density of states and band structure, are central to materials science and condensed matter physics research. The calculation of these properties heavily relies on a key physical quantity called the Green's function. However, when using traditional classical calculation methods to handle strongly correlated systems, the computational complexity explodes exponentially due to the exponential growth of the system's Hilbert space with the number of particles, making efficient and accurate solutions difficult.

[0003] To overcome the bottlenecks of classical computing, researchers have turned to quantum simulation methods. Existing approaches mainly include quantum simulations based on Trotter decomposition and variational quantum algorithms. Trotter decomposition-based methods utilize time evolution operators... The simulation is achieved by decomposing it into the product of a series of short-time evolving sub-circuits, where... Let be the Hamiltonian of the system. However, this method requires a significant increase in quantum circuit depth and total simulation time. This is proportional to the coherence time of current noisy medium-sized quantum processors, making it difficult to obtain sufficiently long evolution data to meet the requirements of Green's function and density of states calculations. On the other hand, variational algorithms, such as variational quantum characteristic solvers, can adapt to the limitations of near-noisy quantum devices, but their main applications focus on the ground state energy of the computational system, making it difficult to directly and efficiently obtain Green's function information that is closely related to real-time dynamic evolution.

[0004] Although the variational fast forward algorithm approximates the short-time evolution unitary operator by diagonalizing it... This algorithm enables long-term quantum simulations with fixed circuit depth, overcoming the limitation of coherence time. However, its current applications are mainly concentrated in areas such as direct dynamic evolution simulation of Hamiltonians and energy eigenvalue estimation. Currently, the variational fast forward algorithm has not been systematically applied to the calculation of Green's functions and density of states in quantum many-body systems, failing to fully leverage its potential advantages in acquiring long-term dynamic evolution data, thus limiting its effectiveness in solving the key problem of the microscopic physical properties of strongly correlated materials.

[0005] Therefore, in order to address the limitations of existing quantum simulation methods in calculating the Green's function and density of states of strongly correlated systems, such as circuit depth constraints or mismatch between application objectives, there is an urgent need for a new technical solution that can effectively integrate the deep compression advantages of the variational fast forward algorithm with the specific requirements of Green's function measurement, so as to achieve efficient and accurate analysis of the physical properties of strongly correlated systems. Summary of the Invention

[0006] The technical problem to be solved by this invention is to address the shortcomings of existing technologies, specifically by providing a method and related equipment for analyzing the physical properties of strongly correlated systems, as detailed below: 1) In a first aspect, the present invention provides a method for analyzing the physical properties of strongly correlated systems, the specific technical solution of which is as follows: Based on the Hamiltonian of the target strongly correlated system, a short-time evolution operator is constructed based on the Hamiltonian. The variational fast forward algorithm is used to approximate the diagonalization of the short-time evolution operator to obtain a parameterized variational quantum circuit containing a unitary transformation part and a diagonal part. The adjustable parameters of the parameterized variable quantum circuit are optimized to minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator, thus obtaining the optimized variable quantum circuit. The optimized variable quantum circuit is embedded as a time evolution submodule into a preset Green's function measurement circuit, and a target quantum circuit with a fixed circuit depth is constructed by scaling the parameters of the diagonal part of the optimized variable quantum circuit. Run the target quantum circuit and obtain the time domain data of the Green's function of the strongly correlated system through quantum measurement; The density of states of a strongly correlated system is calculated by performing a frequency domain transformation on the Green's function data in the time domain. Based on the density of states, the physical properties of strongly correlated systems are analyzed.

[0007] The beneficial effects of the physical property analysis method for strongly correlated systems provided by this invention are as follows: First, by employing the variational fast forward algorithm to approximately diagonalize the short-time evolution operator and optimizing the parameterized variational quantum circuit, the long-term evolution process was successfully compressed into a fixed-depth quantum circuit. This approach directly overcomes the inherent defect of linear growth in circuit depth with simulation time in Trotter decomposition-based quantum simulation methods, making it possible to obtain Green's function data for longer time windows on near-noise quantum processors with finite coherence times, breaking through the time limitations of traditional simulations. Second, by embedding the optimized variational quantum circuit as a time evolution submodule into a pre-defined Green's function measurement circuit, and utilizing parameter scaling of the diagonal portion to construct a target quantum circuit of fixed depth, the variational fast forward algorithm was directly combined with the needs of multibody spectral function calculation. This not only leverages the advantages of the variational fast forward algorithm in long-term evolution simulations but also precisely extends its application scope from general dynamic simulations and energy spectrum estimation to the calculation of Green's function and density of states—core physical quantities in the analysis of strongly correlated systems—solving the deficiency of existing variational algorithms in efficiently obtaining dynamic correlation functions. Ultimately, based on the obtained density of states analysis, the physical properties of strongly correlated systems are obtained, providing effective data from quantum simulations for understanding the electronic structure, conductivity, and phase transition behavior of materials. This entire approach offers a practical new path for efficiently and accurately analyzing the physical properties of strongly correlated systems on noisy quantum devices.

[0008] Based on the above scheme, the physical property analysis method of a strongly correlated system of the present invention can be further improved as follows.

[0009] Furthermore, the cost function is a local Hilbert-Schmidt test cost function, which is used to measure the proximity of the parameterized variable quantum circuit to the short-time evolution operator. Optimizing the adjustable parameters of the parameterized variable quantum circuit to minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator yields the optimized variable quantum circuit. This includes iteratively adjusting the adjustable parameters of the parameterized variable quantum circuit using a classical optimizer to minimize the value of the local Hilbert-Schmidt test cost function, thus obtaining the optimized variable quantum circuit.

[0010] The beneficial effects of adopting the above-mentioned further scheme are as follows: Using a local Hilbert-Schmidt test cost function to measure the approximation between the parameterized variable quantum circuit and the short-time evolution operator, and iteratively optimizing through a classical optimizer, this scheme provides a clear and quantum-measurable optimization objective. This effectively guides the parameter training process of the variational fast forward algorithm, ensuring that the optimized variable quantum circuit can approximate the short-time evolution operator with high precision, laying a reliable foundation for the subsequent construction of a high-fidelity fixed-depth time evolution submodule. Furthermore, a frequency domain transformation is performed on the Green's function data in the time domain to calculate the density of states of the strongly correlated system. This includes performing a Laplace transform or Fourier transform on the Green's function data in the time domain, introducing a positive broadening parameter in the transform, taking the imaginary part of the integral operation result, and calculating the density of states of the strongly correlated system.

[0011] The beneficial effects of adopting the above-mentioned further scheme are as follows: Introducing a positive broadening parameter into the frequency domain transform enhances the stability of the numerical calculation and the rationality of the physical results. The broadening parameter ensures the integral convergence of the transformation process from Green's function data with a finite time length to the frequency domain density of states, while simultaneously simulating the finite energy resolution or quasi-particle lifetime effect of real physical systems. This results in smoother, more physically realistic density of states spectra, helping to extract clear energy spectrum features from potentially noisy quantum measurement data.

[0012] Furthermore, based on the density of states, the physical properties of strongly correlated systems are analyzed, including: using the energy distribution information of electronic states reflected by the density of states to analyze the electronic structure and dynamic behavior of strongly correlated systems within a specific energy range, and evaluating or determining the electrical conductivity, magnetic phase, or superconducting phase of strongly correlated systems based on the electronic structure and dynamic behavior.

[0013] The beneficial effects of adopting the above-mentioned further scheme are: by analyzing the electronic structure and dynamic behavior through the electronic state energy distribution information reflected by the density of states, and thus evaluating macroscopic physical properties, this scheme establishes a direct bridge from microscopic quantum simulation data to macroscopic property judgment. Features in the density of states spectrum, such as band gap, peak position, and spectral weight, are directly related to various physical properties of the system, enabling quantum computing-based results to be used to specifically predict or explain the physical behavior of strongly correlated materials in specific phases, thus completing the closed loop from computation to physical interpretation.

[0014] 2) In a second aspect, the present invention also provides a physical property analysis system for strongly correlated systems, the specific technical solution of which is as follows: It includes a diagonalization module, an optimization module, a target quantum circuit construction module, a data acquisition module, a frequency domain transformation module, and a physical property analysis module; The diagonalization module is constructed to: construct a short-time evolution operator based on the Hamiltonian of the target strongly correlated system, and use the variational fast forward algorithm to approximate the diagonalization of the short-time evolution operator to obtain a parameterized variational quantum circuit containing a unitary transformation part and a diagonal part. The optimization module is used to: optimize the adjustable parameters of the parameterized variable quantum circuit, minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator, and obtain the optimized variable quantum circuit. The target quantum circuit construction module is used to: embed the optimized variable quantum circuit as a time evolution submodule into the preset Green's function measurement circuit, and construct a target quantum circuit with a fixed circuit depth by scaling the parameters of the diagonal part of the optimized variable quantum circuit. The data acquisition module is used to: run the target quantum circuit and acquire the time domain data of the Green's function of the strongly correlated system through quantum measurement; The frequency domain transformation module is used to: perform frequency domain transformation on the Green's function data in the time domain, and calculate the density of states of the strongly correlated system; The physical property analysis module is used to analyze the physical properties of strongly correlated systems based on the density of states.

[0015] Based on the above scheme, the physical property analysis system of a strongly correlated system of the present invention can be further improved as follows.

[0016] Furthermore, the cost function is a local Hilbert-Schmidt test cost function, which is used to measure the proximity of the parameterized variable quantum circuit to the short-time evolution operator; The optimization module is specifically used to iteratively adjust the adjustable parameters of the parameterized variable quantum circuit using a classical optimizer, so as to minimize the value of the local Hilbert-Schmidt test cost function and obtain the optimized variable quantum circuit.

[0017] Furthermore, the frequency domain transformation module is specifically used to: perform Laplace transform or Fourier transform on the Green's function data in the time domain, introduce a positive broadening parameter in the transform, take the imaginary part of the result of the integral operation and perform calculation to obtain the density of states of the strongly correlated system.

[0018] Furthermore, the physical property analysis module is specifically used to: analyze the electronic structure and dynamic behavior of strongly correlated systems within a specific energy range by utilizing the energy distribution information of electronic states reflected by the density of states, and evaluate or determine the electrical conductivity, magnetic phase, or superconducting phase of strongly correlated systems based on the electronic structure and dynamic behavior.

[0019] 3) In a third aspect, the present invention also provides an electronic device, the electronic device including a processor coupled to a memory, the memory storing at least one computer program, the at least one computer program being loaded and executed by the processor, so as to enable the electronic device to implement the physical property analysis method of any of the above-mentioned strongly correlated systems.

[0020] 4) In a fourth aspect, the present invention also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the physical property analysis method for any of the strongly correlated systems described above.

[0021] It should be noted that the beneficial effects of the technical solutions of the second to fourth aspects of the present invention and their corresponding possible implementations can be found in the above description of the technical effects of the first aspect and its corresponding possible implementations, and will not be repeated here. Attached Figure Description

[0022] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments of the present invention will be briefly introduced below: Figure 1 This is a flowchart illustrating a method for analyzing the physical properties of a strongly correlated system according to an embodiment of the present invention. Figure 2 for A schematic diagram of the structure; Figure 3 for A schematic diagram of the structure; Figure 4 For uniform superposition states respectively in and Fidelity between evolved quantum states; Figure 5 For the final result Comparative data; Figure 6 This is a schematic diagram of the structure of a physical property analysis system for a strongly correlated system according to an embodiment of the present invention. Detailed Implementation

[0023] The principles and features of the present invention are described below. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0024] The technical solution of the present invention and how the technical solution of the present invention solves the above-mentioned technical problems are described in detail below with specific embodiments. These specific embodiments can be combined with each other, and the same or similar concepts or processes may not be described again in some embodiments. The embodiments of the present invention will now be described with reference to the accompanying drawings.

[0025] like Figure 1 As shown in the figure, a method for analyzing the physical properties of a strongly correlated system according to an embodiment of the present invention includes the following steps: S1. Construct a short-time evolution operator based on the Hamiltonian of the target strongly correlated system, and use the variational fast forward algorithm to approximate the diagonalization of the short-time evolution operator to obtain a parameterized variational quantum circuit containing a unitary transformation part and a diagonal part. Specifically, a short-time evolution operator based on the Hamiltonian of the strongly correlated target system is constructed, and the specific implementation process is as follows: S10. Clarify the specific form of the Hamiltonian of the target strongly correlated system. The Hamiltonian is the energy operator describing the interactions between all particles in the system, and it completely determines the dynamic and static properties of the system. For a given strongly correlated system, such as the Hubbard model or the Heisenberg model, its Hamiltonian... It has a specific analytical expression, which includes physical parameters such as hopping intensity, in-situ Coulomb repulsion energy, and chemical potential. For example, for a two-lattice Hubbard model, its Hamiltonian can be written as... In this formula, and They are located at grid points upper spin is The fermion creation and annihilation operators, It is the particle number operator. It is the transition amplitude. It is the strength of in-situ interaction. It is a chemical potential. Therefore, the starting point for the construction work is to obtain a completely definite mathematical expression for the Hamiltonian.

[0026] S11. Mapping the abstract Hamiltonian onto the framework of qubits: Quantum processors are composed of qubits, therefore the Hamiltonian needs to be mapped... The portion of the Hamiltonian typically represented by fermions or spin operators is converted into a tensor product form of Pauli operators acting on multiple qubits. This conversion is usually achieved through mapping methods such as the Jordan-Wigner transform or the Bravyi-Kitaev transform. After the mapping, the Hamiltonian is rewritten in the standard form of the qubit Hamiltonian: In this formula, It is a summation index. The coefficients are real numbers. It is a Pauli string, that is, a tensor product of multiple Pauli operators, for example For each qubit index ,have , It is the identity matrix. It is a Pauli matrix. For example, a specific... It may manifest as This means acting on the second qubit. The gate operates on the third qubit. Gate, the role of the remaining qubits The gate. The output of this step is a Hamiltonian that is entirely described by qubit operators.

[0027] S12. Select a small time step. Time step It is a positive real number parameter that needs to be preset, and its selection requires balancing two aspects: on the one hand, It needs to be small enough to ensure that the short-time evolution operators constructed based on this step size are small enough. With unit operator The deviation is small, which makes it easier for the optimization process to converge to a good solution when the variational fast forward algorithm is used to approximate diagonalization. On the other hand, The value also determines the final total simulation time. The granularity, of which is a positive integer. generally, The value of is in arrive The order of magnitude (in reciprocal energy units) can be determined through classical numerical simulations or by estimation based on the scale of the Hamiltonian's eigenenergy spectrum. For example, if the Hamiltonian... The range of eigenvalues ​​is approximately Then you can choose , making The range of eigenvalues ​​falls within Inside.

[0028] S13, Although short-time evolution operators are mathematically derived from... While precisely defined, at the quantum circuit level, it requires constructing a specific sequence of quantum gates to implement it. This is due to the Hamiltonian... It typically contains many non-commutative terms, and is directly implemented. It is difficult. The most common method is to use Suzuki-Trotter decomposition, which approximates the complex exponential operator as a product of exponential operators corresponding to the subterms of the Hamiltonian. For a Hamiltonian already mapped to the summation of Pauli strings... An approximate circuit can be constructed using a first-order Suzuki-Trotter decomposition: In this approximation This represents the operator implemented by the final constructed quantum circuit. Indicates a certain order (e.g., index). The product of (in order). Each basic component in the circuit. Corresponding to a Pauli string The exponential form of it can be realized through combinations of basic quantum gates. For example, if ,but This is equivalent to applying a rotation angle of on the first qubit. of Revolving door. If ,accomplish This requires a combination of entanglement gates and single-bit rotation gates. In this way, the Hamiltonian is... Each Pauli string item By generating the corresponding quantum circuits and sequentially assembling all these quantum circuits, a complete quantum circuit is formed, which enables the operation of short-time evolution operators. An approximation of, namely The constructed circuit is the target module that the variational fast forward algorithm will approximate diagonalize in subsequent steps. The output of the entire construction process is a well-defined and executable quantum circuit that represents the short-time evolution operator and serves as a fixed comparison benchmark for the subsequent variational optimization stage.

[0029] Among them, the short-time evolution operator is a fundamental mathematical object describing the dynamic behavior of a quantum system over an extremely short time. In this invention, the short-time evolution operator specifically refers to the time evolution operator corresponding to a tiny time step, generated by the Hamiltonian of the strongly correlated target system. Its mathematical form is: ,in It is the Hamiltonian of the system. It is a selected, very small positive real time step. The operator's function is to evolve a quantum state forward from a certain initial time step. In such a short time, constructing the quantum circuit representation of this operator means finding a series of quantum logic gates such that the overall effect of these gates acting sequentially on the qubits is approximately equivalent to a mathematical operator, within an acceptable margin of error. Its function.

[0030] The variational fast forward algorithm is used to approximate the diagonalization of the short-time evolution operator, resulting in a parameterized variational quantum circuit containing both a unitary transform part and a diagonal part. The specific implementation process is as follows: S14. Based on the principle of the variational fast forward algorithm, it is necessary to clearly construct... and The circuit template. For the unitary transform part. Quantum circuits typically employ a layered structure. For example, a hardware-friendly entanglement layer can be defined. This layer contains fixed two-qubit entanglement gates (such as CNOT gates or CZ gates) and trainable single-qubit rotation gates (such as... and (Door). The whole Then by Layers like this It is constructed by concatenation, and additional single-bit rotation gates may be added at the beginning or end to increase expressive power. Expressed as follows: ,in It is the first Layer parameters, It is a collection of all layer parameters. This represents an additional single-bit rotation gate. For the diagonal portion... A common option is to limit it to include only Pauli. The tensor product term of the operator, because The gates are diagonal under the computational basis. Finite orders can be obtained. ,For example until it includes Volume interaction term. Here It is the set of all rotation angles. It refers to the number of qubits. The choice of circuit template needs to balance expressive power and feasibility on real quantum hardware.

[0031] S15. Before starting optimization, the parameter vector needs to be... and Set initial values. Parameters It can be initialized randomly, for example from a uniform distribution. The value of each rotation angle is randomly selected from the parameters. Initialization can consider short-time evolution operators. The minuteness. Due to You can try to Initialize to Multiply the diagonal terms (under the computational basis) by The relevant values ​​can be initialized using small random numbers. The initialization strategy affects the speed and effectiveness of optimization convergence.

[0032] S16. A measurable quantity needs to be defined for quantification. and The difference. This invention employs a local Hilbert-Schmidt test cost function. The function is defined as follows: ,in It is the total number of qubits. This is the entanglement fidelity term that can be estimated using a short circuit involving auxiliary qubits. To estimate this cost function on a quantum processor, it is necessary to perform calculations for each pair of entanglement qubits. Prepare a specific quantum circuit (i.e., a local Hilbert-Schmidt test circuit), run it multiple times and measure the auxiliary qubits, and statistically analyze the specific outputs in the measurement results (such as...). Calculate the probability of the state. And thus obtain The current value of the quantum circuit. This evaluation process converts the output of the quantum circuit into a scalar value to guide classical optimization.

[0033] S17. Adjusting the adjustable parameters of the parameterized variational quantum circuit using a classical optimizer is an iterative process. The classical optimizer (such as gradient descent, conjugate gradient, or adaptive moment estimation) evaluates the cost function value obtained in the previous step. The direction and magnitude of the parameter adjustment are calculated. Then, new parameter values ​​are generated. and Next, the parameterized variable quantum circuit is configured with the new parameters. The new cost function value is then evaluated again on the quantum processor. This process is repeated cyclically, with the goal of continuously reducing the cost function. The value of .

[0034] S18. After each iteration, check whether the cost function value is lower than a preset tolerance threshold. Or, check if the number of iterations has reached the preset upper limit. If the convergence condition is met, stop the optimization and record the current optimal parameters as... and At this point, the parameterized variable quantum circuit It is considered a short-time evolution operator. A high-quality approximate diagonalized version. This optimized variational quantum circuit includes a unitary transform part with a fixed structure. and diagonal part The circuit obtained in this step is the result of this process. It will be embedded as a reliable time evolution submodule into the subsequent Green's function measurement circuit.

[0035] The unitary transform part refers to the component responsible for basis transformation in the parameterized variational quantum circuit. Within the framework of the variational fast forward algorithm, it is assumed that the short-time evolution operator... It can be approximated as diagonal, that is, there exists a unitary transformation that will This is transformed into a representation primarily composed of diagonal gates. Unitary transformation part. This is the parameterized quantum circuit implementation of the unitary transformation. It represents a parameter vector. Its function is to conceptually... The eigenstates of the eigenstates are rotated onto the computational basis of the quantum processor (i.e., the qubits). and (The space spanned by states). Since the real eigenstates and transformations are usually unknown and difficult to construct directly, they are constructed through a space with adjustable parameters. quantum circuits We use variational approximation to approximate this ideal transformation. The design of the unitary transform circuit directly affects its expressive power and the accuracy of the final approximation.

[0036] The diagonal part refers to the component that simulates the diagonal matrix in a parameterized variable quantum circuit. In diagonalization decomposition, the diagonal part... A parameterized quantum circuit implementation corresponding to a diagonal matrix, where This represents another parameter vector. In quantum circuits, It typically consists of a series of commutative rotation gates acting on different qubits, for example... A revolving door. Its mathematical form can be represented as: Or, more generally, a truncated summation form containing diagonal terms of multiple qubits. The diagonal part... The core feature is that their gate operations are mutually commutative, which makes powers This can be easily achieved by scaling the rotation angle, i.e. This characteristic is the root cause of achieving fixed-depth time evolution.

[0037] In this context, parameterized variational quantum circuits refer to complete quantum circuit models with tunable parameters constructed to approximate short-time evolution operators. Specifically, in the variational fast forward algorithm, parameterized variational quantum circuits... It is constructed into a specific form combining the unitary transformation part and the diagonal part: .in yes Hermitian conjugate (i.e., inverse transformation). This structure intends to explicitly express an approximately diagonalization process: first through... Transform to an approximate intrinsic basis, and then in which... Apply phase rotation, and finally pass Transform back to the original basis. Parameterized variable quantum circuit. The design goal is to optimize and adjust the parameter vector. and To make it as close as possible in function to the target short-time evolution operator. .

[0038] S2. Optimize the adjustable parameters of the parameterized variable quantum circuit to minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator, and obtain the optimized variable quantum circuit. The cost function is a local Hilbert-Schmidt test cost function, which measures the proximity of the parameterized variable quantum circuit to the short-time evolution operator. In S2, the adjustable parameters of the parameterized variable quantum circuit are optimized to minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator, resulting in the optimized variable quantum circuit. This includes iteratively adjusting the adjustable parameters of the parameterized variable quantum circuit using a classical optimizer to minimize the value of the local Hilbert-Schmidt test cost function, thus obtaining the optimized variable quantum circuit. The specific implementation process is as follows: S20. Construct a quantum measurement circuit for evaluating the local Hilbert-Schmidt test cost function, which is defined as follows: To estimate this value on a quantum processor, it is necessary to perform calculations for each specific pair of target operators. (i.e., short-time evolution operators) and variational operators under the current parameters and the index of each quantum bit A dedicated test circuit needs to be constructed. This test circuit requires an auxiliary qubit and two sets of circuits, each containing... A system with 100 qubits. For a fixed... Entanglement fidelity item The estimation circuit operates as follows: First, the auxiliary qubit is connected to the first... The system's qubits are prepared into a specific entangled state; then, in one of them... Acting target operator on a quantum bit system In another Variational operators to be evaluated on a quantum bit system Then, a series of controlled operations and inverse entanglement preparation operations are performed; finally, projection measurements are taken on the auxiliary qubit. By running this circuit multiple times and statistically analyzing the state of the auxiliary qubit... Frequency of states (probability) ), can be estimated The design of this circuit makes The value can be directly reflected by the observable probability.

[0039] S21, Apply the previous step to all from arrive Constructed Several different test circuits are run sequentially or in parallel on the quantum processor. Each circuit needs to be run a sufficient number of times (called the sampling number). To obtain the probability Stable statistical estimates for the . For the . A circuit records the measured auxiliary qubits. Number of times and calculate And thus obtain In obtaining all indivual After obtaining the estimated value, substitute it into the formula for the local Hilbert-Schmidt test cost function for calculation: This calculation process is typically performed on a classical computer. The result is... It's a scalar value that quantifies the current parameter setting. Below, parameterized variable quantum circuit With the target short-time evolution operator The degree of closeness between them. The smaller the value, the higher the degree of closeness.

[0040] S22. The classical optimizer receives the current cost function value from the quantum-classical interface. and the current adjustable parameter vector Internally, the optimizer determines how to modify these parameters to reduce the cost function value in the next step, based on its algorithmic logic. For example, if gradient descent is used, the optimizer might need to estimate the gradient of the cost function with respect to each parameter. This can be done by methods such as parameter shifting rules, or by running some specially configured quantum circuits to estimate the gradient components. If a gradient-free optimizer (such as the Nelder-Mead method) is used, a simple model of the parameter space might be constructed based on the cost function values ​​of the current and previous iterations, and new parameter trial points might be proposed accordingly. Regardless of the strategy used, the optimizer's output is a new set of adjusted, tunable parameter values, denoted as . .

[0041] S23. Apply the new parameters generated by the classic optimizer. The control software sent to the quantum processing system is used to reconfigure the parameterized variable quantum circuit. Then, the process jumps back to S20, constructs a new test circuit using the new parameters, and begins a new round of measurement, calculation, and optimization. This cycle of "quantum measurement evaluation -> classical optimization update" repeats continuously. At the end of each iteration, it is necessary to check whether the preset convergence conditions are met. There are generally two types of convergence conditions: one is the cost function value. Falling below a certain threshold ,For example Another scenario is when the improvement in the cost function value is very small in consecutive iterations, or when the preset maximum number of iterations is reached. Convergence criteria are determined on a classical computer.

[0042] S24. Once the above convergence condition is met, the iterative optimization process terminates. At this point, the parameters used in the current iteration are... Determined as the optimal parameter, denoted as Accordingly, parameterized variable quantum circuits After applying these optimal parameters, it becomes an optimized variable quantum circuit, the specific form of which is: This optimized variational quantum circuit is considered to have successfully approximated the diagonalized short-time evolution operator with high fidelity. Its circuit structure ( and The template is fixed, while the parameters are... It is definitive and can be used to subsequently construct fixed-depth circuits that evolve over arbitrarily long periods.

[0043] In this context, a classical optimizer refers to an algorithm running on a classical computer that seeks parameter values ​​that minimize the objective function. Within the framework of variational quantum algorithms, the classical optimizer processes the measurement data (i.e., cost function values) returned from the quantum processor and calculates how the adjustable parameters of the parameterized variational quantum circuit should be adjusted next, based on a specific optimization strategy. Common classical optimizers include gradient descent, conjugate gradient, Nelder-Mead simplex, and adaptive moment estimation. These optimizers do not need to understand the internal physical details of the quantum circuit; instead, they treat it as a "black box" function that takes a parameter vector as input and outputs a scalar cost function value. The optimizer's task is to efficiently explore the parameter space, guiding the parameterized variational quantum circuit to approximate the target short-time evolution operator.

[0044] In this context, the adjustable parameters of a parameterized variational quantum circuit refer to the set of variables introduced during the circuit's construction, whose values ​​can continuously vary within a certain range. These variables typically correspond to the rotation angle of a rotating gate in a quantum circuit. Specifically, in the variational fast forward algorithm, the adjustable parameters mainly consist of two parts: one part is the unitary transform part. parameter vector in The other part is the diagonal part parameter vector in .in and These represent the total number of parameters in the two categories, respectively. These parameters are assigned initial values ​​(e.g., randomized) at the start of optimization and are iteratively updated by the classical optimizer during the optimization process. The adjustable parameters of the parameterized variable quantum circuit are a crucial bridge connecting classical optimization logic and quantum hardware execution; their final optimized values ​​determine... right Approximate mass.

[0045] S3. The optimized variable quantum circuit is embedded as a time evolution submodule into the preset Green's function measurement circuit. By scaling the parameters of the diagonal part of the optimized variable quantum circuit, a target quantum circuit with a fixed circuit depth is constructed. The specific implementation process is as follows: S30. Calculate the delayed Green's function. It can be converted into calculating a called The quantity can be estimated using a specific quantum circuit. This circuit typically requires an auxiliary qubit and several system qubits. The circuit operation sequence includes: initializing all qubits; applying a Hadamard gate to the auxiliary qubit; and executing a controlled... Operation, among which Encoded AND operator and Related transformations; then the time evolution of the system is executed. Next, another controlled operation may be performed; finally, another Hadamard gate is applied to the auxiliary qubit and a measurement is taken. In the measurement result, the auxiliary qubit is in... probability of state and the quantity to be determined Directly related, the formula is This circuit structure is the pre-designed Green's function measurement circuit, in which... Some of these are modules that need to be replaced.

[0046] S31. Embed the optimized variable quantum circuit to replace the time evolution part in the preset circuit. In the preset Green's function measurement circuit, find the part originally used to implement... The line segment (i.e.) (Partial). Remove this entire section of circuitry and insert the optimized variable quantum circuit in the same location. However, there is a key point here: It is approximately a short time evolution operators What is needed is any time. evolution operators Therefore, direct insertion Only calculation The Green's function at any time. To calculate the Green's function at arbitrary time... We need to initiate the next parameter scaling operation.

[0047] S32. According to the mathematical principles of the variational fast forward algorithm, if and So for a long time There is an approximate equation More importantly, because It consists of commutative gates, and its powers can be simplified to scaling of parameters: Therefore, in order to implement the Green's function measurement circuit... It is not necessary to The entire circuit is repeatedly connected in series. (that would lead to linear growth in depth) (times), while only requiring a single embedding. The circuit structure, but its internal diagonal part All parameters Multiply by a scaling factor That is, construct a new diagonal part. So, for long-term use The module of evolution is .this circuit gate sequence and They are exactly the same, the only difference being that the rotation angle of all the revolving doors varies depending on their position in the room. The position in the middle has been enlarged proportionally. Times. Therefore, its circuit depth is times that of... Completely consistent; it is fixed.

[0048] S33. Scale the parameters of the long-term evolution module obtained in the previous step. Completely replace it in the preset Green's function measurement circuit. The position. At this time, all other parts in the preset circuit (such as initial state preparation, controlled state) are in the correct position. Operations, final-state measurements, etc., remain unchanged. After assembly, a specific evolution time is obtained. The target quantum circuit. Due to its time evolution submodule. It has a fixed depth, and the depth of other components in the circuit is the same as... Since it is irrelevant, the total depth of the entire target quantum circuit is also fixed and does not change with... It increases with the increase of the target evolution time. No circuit redesign or deep optimization is required; only the corresponding scaling factor needs to be calculated on a classic computer. Then use By configuring the diagonal parameters within the same circuit template, you can generate the corresponding time-based parameters. The target quantum circuits. This series of circuits shares the same gate sequence and depth, differing only in parameters, thus achieving the goal of efficiently computing the Green's function in the long time domain.

[0049] The time evolution submodule refers to the module used in this invention to replace the precise time evolution operator. A quantum circuit component based on the variational fast forward algorithm. Specifically, it is an optimized variational quantum circuit. This module is designed to approximate the system starting from the initial moment and lasting for a tiny period of time. The evolution of [the process]. Its key characteristic is that when a longer evolutionary time needs to be simulated... Instead of repeating or cascading multiple modules that would cause the circuit depth to grow linearly, this can be achieved through simple scaling of internal parameters, thus keeping the physical quantum gate depth of the entire submodule constant when it acts as a long-term evolution operator.

[0050] The preset Green's function measurement circuit refers to the circuit designed for calculating specific Green's function matrix elements. The pre-designed quantum circuit framework, based on existing quantum algorithms, can transform the real-time expression of the Green's function into a form that can be estimated through the probability of measuring the final state of the quantum circuit. A typical pre-designed circuit might include the following components: circuitry for preparing specific initial states (such as uniform superposition states), and circuitry for encoding fermionic operators. or The circuit block that performs the function (called (partial), and used for execution time evolution. The line block (called (Partial). This invention does not fundamentally change the overall structure of the preset circuit, but focuses on the time evolution part (i.e. (Partial) replacements and enhancements.

[0051] Among them, a target quantum circuit with a fixed circuit depth refers to: a final assembled quantum circuit used to calculate a specific time. The complete quantum circuit corresponding to the Green's function. Its "fixed circuit depth" characteristic specifically refers to: regardless of the computation time... The length of the circuit does not change significantly; the number of quantum logic gate layers (i.e., depth) remains essentially constant. This growth is achieved by embedding an optimized variable quantum circuit as a time-evolution submodule and leveraging its parameter scaling properties, in stark contrast to the linear growth of circuit depth over time based on traditional Trotter decomposition. The target quantum circuit is a final program that can be directly compiled and run on a quantum processor.

[0052] S4. Run the target quantum circuit and obtain the Green's function of the strongly correlated system in the time domain through quantum measurement. The specific implementation process is as follows: S40, for a selected evolution time The target quantum circuit has already been constructed in the preceding steps, and its circuit depth is fixed. Before execution, the circuit described using high-level quantum gates needs to be converted into a set of quantum gates natively supported by the specific quantum processor using a compilation toolchain provided by the quantum hardware vendor. The topological connectivity constraints of its qubits must also be considered to complete the mapping and routing of the qubits. After compilation, the generated executable instructions are loaded into the quantum processor's control system. Simultaneously, an important operating parameter needs to be set: the number of samples, or the number of measurements. This parameter determines the number of repetitions required for projective measurements of the circuit's final state. The larger the value, the higher the accuracy of the statistical estimate, but the longer the total running time required.

[0053] S41. Based on the received instructions, the quantum processor initializes all qubits to their default states, and then sequentially executes all quantum logic gate operations in the circuit. At the end of the circuit, a projection measurement is performed on one or more specified qubits, collapsing their states to computational basis vectors. or Above. Due to the probabilistic nature of quantum measurements, a single run yields a definite bit string result. For example, in a quantum world containing one auxiliary qubit and... In a circuit with 1000 qubits, each measurement yields a value of length 1000. The bit string. This process is repeated independently. The quantum processor's control system, or a connected classical computer, records the result of each measurement, ultimately compiling the data into a sequence of length [length missing]. A list of measurement results. Each result is a binary string in the form of "0101...".

[0054] S42. Based on the design of the preset Green's function measurement circuit, the Green's function... The estimated value is the same as the measurement result of a specific qubit (usually an auxiliary qubit) in the circuit. probability Directly related. Specifically, there exists a conversion formula. ,and And with The real and imaginary parts of the probability have a linear relationship (depending on the circuit design). Therefore, the core of data processing is calculating the probability. The estimated value The specific method is: From all the collected... In a given set of measured bit strings, count the number of bit strings where the bits corresponding to the auxiliary qubit positions are '0', denoted as . Then calculate. .this It is the auxiliary quantum bit measurement The frequency is a measure of the true probability. An unbiased estimate.

[0055] S43, take the result from the previous step Substituting the pre-defined mathematical relationship derived from the circuit, the Green's function is calculated. The estimated value , The real or imaginary part is proportional to Therefore, the calculation process is as follows: First calculate... Then, based on the specific proportional coefficient relationship, we obtain... If the circuit design allows the output to directly correspond to the real part of the Green's function. ,So It's just a real number estimate; if you need to obtain both the real and imaginary parts simultaneously, you might need to run two slightly different circuit variants (e.g., changing the phase). Ultimately, you get a numerical value. It is a theoretical value The approximation with statistical noise.

[0056] S44, S40 to S43 above are for a single evolutionary time. The operation involves... To obtain a graph of the Green's function over time, a time series needs to be pre-defined. ,in , This represents the maximum simulation time. For each time point in the sequence... Repeat the entire process: build the corresponding The target quantum circuit (by scaling diagonal parameters), compilation, and execution. Calculations were performed to obtain... When for all After all the time points have been calculated, a complete set of time-domain data points is obtained. This set of data constitutes a function. In the time interval The discrete approximation on the frequency domain is the foundation for subsequent frequency domain analysis. Because each The corresponding target quantum circuit depth is fixed, therefore even Very large The data is very large, and obtaining this data point is theoretically feasible, breaking through the bottleneck of traditional methods that cannot simulate long-term performances due to circuit depth limitations.

[0057] In strongly correlated systems, the Green's function is the core correlation function describing the generation, propagation, and annihilation of particle or quasi-particle excitations. Under zero-temperature conditions, the retarded Green's function, commonly used to describe single-particle excitations, is defined as follows: In this formula, and It is a grid or other degree of freedom index. and These are the fermion annihilation operator and the production operator as depicted in Heisenberg's painting. Expressing opposition to exchanging children, It is a step function. This represents the expected value of the system in its ground state. This function contains key physical information such as the system's energy spectrum, density of states, and transport properties.

[0058] In this context, the Green's function in the time domain of a strongly correlated system refers to: the delayed Green's function. At a series of discrete time points The set of numerical estimates on, where Usually, fixed and examine With evolution time The changing relationships. Time-domain data is a set of ordered pairs. ,in, It is the selected evolutionary time. The theoretical value was obtained through quantum computing and measurement. This is an approximate estimate. Obtaining this series of data is a prerequisite for subsequent frequency domain transformation to calculate the density of states.

[0059] S5. Perform a frequency domain transformation on the Green's function data in the time domain to calculate the density of states of the strongly correlated system. Specifically, perform a Laplace transform or Fourier transform on the Green's function data in the time domain, introduce a positive broadening parameter in the transform, take the imaginary part of the integral result and calculate it to obtain the density of states of the strongly correlated system. The specific implementation process is as follows: S50, the input data is a series of discrete data points obtained by running the target quantum circuit and measuring them, denoted as... .in, , It is the maximum simulation time. Are they time points with equal or unequal intervals? It is the Green's function that delays the corresponding time. The estimated value. Typically, to ensure the numerical stability of the transformation, data quality needs to be checked, and data preprocessing may be performed, for example, by utilizing the symmetry of the Green's function (e.g., ...). This can be used to supplement or constrain data, or to smooth out obvious statistical outliers.

[0060] S51. The frequency range of the density of states to be calculated needs to be determined. and resolution Thus defining the frequency grid points ,in At the same time, based on system characteristics and data time windows... Select a specific positive value for the broadening parameter One rule of thumb is... It should be slightly larger than To suppress Gibbs oscillations caused by finite-time truncation, a numerical integration algorithm is then chosen to compute the continuous integral. Discrete approximations are obtained. Commonly used methods include the trapezoidal rule and Simpson's rule. Due to the integrator kernel... It is oscillatory and decaying, for larger ones and Simple numerical integration methods are usually sufficient.

[0061] S52, For each frequency point on the grid Calculate complex values approximation The calculation process involves discrete summation: In this formula, These are the weighting coefficients for numerical integration methods (for example, for the trapezoidal rule, the weights at the two endpoints are 0.5, and the weights at the midpoints are 1). It is the time step. Summation from Accumulated to This covers the entire effective time window. This operation is essentially an approximation of the Laplace transform (or can be viewed as a Fourier transform with a decay factor). For each... This calculation outputs a complex number, whose real and imaginary parts both contain physical information.

[0062] S53. According to the definition of local density of states, for lattice points In the case of density of states The negative value of the imaginary part of the Green's function in the frequency domain is given by: Therefore, for each frequency point Take the complex number obtained in the previous step. The imaginary part, i.e. Then multiply it by a coefficient. Thus, the estimated value of the density of states at that frequency point is obtained. : because It originates from quantum measurements and numerical approximations that contain statistical noise. It is also an approximation.

[0063] S54. Calculate all the points obtained in the previous step. Connecting them together constitutes the local density of states of a strongly correlated system at lattice point 1 as a function of energy. The graph shows the distribution of electronic states in energy space. Further analysis of the spectral characteristics is possible, such as the peak position corresponding to the quasiparticle excitation energy of the system, and the peak width relating to the quasiparticle lifetime or broadening parameters. Correlation, spectral weights reflect the characteristics of the states. The entire transformation process successfully converts the time-varying correlation function data obtained from the quantum processor into the energy-varying density of states spectrum, which is more familiar to physicists, providing a direct basis for analyzing the electronic structure of the system. It should be noted that the quality of the final spectral lines depends simultaneously on the precision and length of the time-domain data obtained by the quantum part, as well as the parameters of the classical transformation part (such as...). The rationality of the choice.

[0064] Here, the positive broadening parameter is a small positive real number introduced in the frequency domain transform integral, usually denoted as . Its main physical and mathematical functions are reflected in two aspects: First, from a mathematical perspective, it enables integrals... Factors in Become a decay factor to ensure that the integrand changes over time. As it approaches infinity, it rapidly decays to zero, thus ensuring numerical convergence of the integral even when using data within a finite time window. Secondly, from a physical perspective... The introduction of this is equivalent to applying a Lorentzian-type broadening to the system's energy levels, with a full width at half maximum (FWHM) of approximately [value missing]. This reflects a real-world reality: any real measurement or finite-lifetime quasiparticle excitation is limited by energy resolution. The value of needs to be carefully selected. If it is too small, it will cause severe oscillations in the numerical integral and be sensitive to finite-time truncation errors. If it is too large, it will over-smooth the spectral features and mask the true physical details.

[0065] S6. Based on the density of states, analyze the physical properties of strongly correlated systems. Specifically, using the energy distribution information of electronic states reflected by the density of states, analyze the electronic structure and dynamic behavior of strongly correlated systems within a specific energy range, and evaluate or determine the electrical conductivity, magnetic phase, or superconducting phase of the strongly correlated system based on the electronic structure and dynamic behavior. The specific implementation process is as follows: S60. Read and visualize the calculated density of states data. The analysis process begins with a classical computer reading the final data file output from the previous steps, which contains a series of frequency-density of states pairs. These data points were plotted as a curve using scientific graphing software, i.e., a density of states spectrum. The horizontal axis represents energy. (usually in eV or hopping intensity) (units), vertical axis is Visualization allows for an intuitive observation of global features such as the overall shape of the spectral lines, peak positions, peak widths, and spectral weight distribution.

[0066] S61. On the visualized spectrum, focus on a specific energy range for detailed analysis. First, the region near the Fermi level. This usually corresponds to the zero energy point. .observe density of states at :if A finite value indicates that the system possesses electronic states at the Fermi level and exhibits metallic characteristics; if Or close to zero, and in a state of... Centered on a limited energy window If the internal state density is consistently low, it indicates that the system has an energy gap. It exhibits characteristics of either an insulator or a semiconductor. The next step is to identify peak structures far from the Fermi level, which may correspond to the upper Hubbard band, lower Hubbard band, hybrid band, or other nontrivial excitations.

[0067] S62. Based on the features identified in S61, inferences are made about the microscopic electronic structure of the system. For example, if it is observed that... A narrow spike centered on this could suggest quasi-particle excitations of near-Fermi liquids in strongly correlated metals. If observed... The density of nearby states is strongly suppressed, and in Two broad peaks appeared nearby ( (This is the in-situ Coulomb energy), which conforms to the typical two-band structure of Mott insulators. The dynamic behavior is reflected by the width and shape of the spectral lines: sharp peaks usually correspond to long-lived quasiparticle excitations, while broad bulges or flat bands may indicate short-lived, highly localized excitations, or the presence of strong inelastic scattering.

[0068] S63. Based on the inferred electronic structure and dynamics picture, evaluate the macroscopic physical phases and properties that the system may exhibit. Regarding conductivity: If the spectrum shows... If a system has a non-zero density of states and a sharp quasiparticle peak, it is very likely to behave as a good conductor; if If it is located within the band gap, then the system is an insulator; if Crossing a narrow region with a very low density of states may result in a half-metallic appearance. Regarding magnetic phases: certain magnetic orders (such as antiferromagnetism) can introduce new split structures or alter the spectral weights of specific energy positions in the density of states spectrum. By comparing theoretical density of states predictions under different magnetic order assumptions, the existence of a certain magnetically ordered phase can be supported or ruled out. Regarding superconducting phases: the density of states of conventional superconductors within the superconducting band gap... Strange peaks appear at the edges. Although the spectrum of strongly correlated superconductors is more complex, The dramatic rearrangement of the nearby density of states and the appearance of the bandgap remain key signals. During analysis, the focus is on finding... The density of states is suppressed (bandgap opening), and in The characteristic of coherent peaks appearing at a certain location.

[0069] S64. Based on all the above analyses, form a conclusive description of the physical properties of this strongly correlated system. For example, the conclusion might be: "For a given parameter ( The two-dimensional Hubbard model, when the particle is half-filled, does not show a finite peak in the density of states at the Fermi level, while... A broad peak was observed, consistent with the expectation of Mott-Hubbard insulators, indicating that the system behaves as a non-conductive insulating phase. The output of this step is a qualitative physical judgment and quantitative characteristic parameters (such as bandgap size). Quasi-particle weights Peak width wait.

[0070] Among them, the density of states, which reflects the energy distribution information of electronic states, refers to the distribution of electronic states by a function. The described, in energy The number of quantum states available for electrons to occupy within a unit energy interval. Specifically, for a given strongly correlated system, the calculated local density of states. It is an energy This is a curve with the x-axis representing the x-axis and the density of states representing the y-axis. Each point on this curve... Directly revealing the energy The probability density of nearby electronic states. Peak regions indicate the presence of a large number of occupyable electronic states within that energy range, typically corresponding to energy bands or discrete energy levels; zeros or minimum values ​​on the curve within a certain energy range may indicate the existence of a band gap. This distribution information is fundamental to understanding the electronic structure of a system.

[0071] Among them, strongly correlated systems include high-temperature superconducting materials, Mott insulators, heavy fermion materials, or transition metal oxides, etc. Other strongly correlated systems may also be selected depending on the circumstances. Specifically: (1) When the target strongly correlated system is a high-temperature superconducting material, the technical solution of this invention provides a systematic path from quantum computing simulation to specific physical property analysis. High-temperature superconducting materials generally refer to material systems such as copper oxides and iron-based compounds that have transition temperatures much higher than those of traditional superconductors, and whose superconducting mechanism is closely related to strong electron correlation effects. Analyzing its physical properties using this invention involves obtaining dynamic information about its microscopic electronic structure through quantum simulation, and then interpreting and evaluating its macroscopic properties related to superconductivity and other competing orders.

[0072] The physical properties of high-temperature superconducting materials refer to the various observable characteristics exhibited by these materials on a macroscopic scale, which are closely related to strong electron correlation. These mainly include, but are not limited to, the properties at a specific critical temperature. The following properties exhibit zero resistance and perfect diamagnetism (i.e., superconductivity); in the normal state (above the superconducting transition temperature), they often exhibit non-Fermi liquid behavior, pseudo-bandgap phenomena, anomalous Hall effects, and various competing orders such as charge density waves, spin density waves, and fringe phases that may coexist with superconductivity. Understanding the microscopic origins of these properties, especially the superconducting pairing mechanism, is one of the core challenges in condensed matter physics. This invention provides a direct means to explore the electronic structure behind these properties by calculating dynamic spectral functions such as local density of states. The specific process is as follows: 1) Constructing an effective model Hamiltonian for high-temperature superconducting materials: The complex crystal structure and electronic interactions of high-temperature superconducting materials are usually simplified to a strongly correlated electron model defined on the crystal lattice. One of the most commonly used models is the two-dimensional Hubbard model or its derivatives (such as...). - (Model). For example, consider a model describing copper oxide CuO. The Hamiltonian of a planar Hubbard model can be written as: In this formula, and They are grid points upper spin is The electron creation and annihilation operators, This represents summing over nearest neighbor grid points. It is the nearest neighbor transition integral. It is the in-situ Coulomb repulsion energy at the lattice point. It is the particle number operator. It is the chemical potential. For different families of high-temperature superconducting materials, the parameter... , and possible next nearest neighbor jumps These need to be determined based on first-principles calculations or experimental fitting. This step determines the physical objective for all subsequent quantum simulations.

[0073] 2) Implement technical solutions S1 to S5 to obtain the local density of states spectrum under this model. Specifically: ① Based on the selected effective model Hamiltonian of high-temperature superconducting materials, construct the quantum circuit representation of its short-time evolution operator. For a given two-dimensional Hubbard model Hamiltonian... The short-time evolution operator is defined as follows: ,in It is a pre-defined, tiny time step. The construction process involves transforming the mathematical operator into an approximate circuit that can be implemented by a sequence of quantum gates. First, the Hamiltonian is transformed using the Jordan-Wigner transform. The mapping is in the form of summation of Pauli tensor products (Pauli strings): Subsequently, a first-order Suzuki-Trotter decomposition is used to approximate the result. ,Right now This approximation means that for each term in the Hamiltonian, we need to... Design a corresponding quantum circuit. For example, for The term, whose circuit is an operation on a specific quantum bit. Rotating gate; for off-diagonal terms involving two qubits, such as This requires the use of a combination of CNOT gates and single-qubit rotation gates for construction. Finally, these sub-circuits are assembled in a defined order to form a complete quantum circuit that realizes the short-time evolution operator for the target. Approximate to .

[0074] ② The variational fast forward algorithm is used to approximate the diagonalization of the constructed short-time evolution operator circuit. The goal of this sub-step is to find a parameterized variational quantum circuit. To make it as close as possible in function. First, the unitary transformation part needs to be designed. and diagonal part The circuit template (i.e., Ansatz). It typically consists of a multi-layered structure containing entangled gates and adjustable parameter rotating gates; Then it consists of a series of interchangeable diagonal doors (such as...) (Rotation) constitutes the composition. Next, all adjustable parameters are initialized. and Then, a quantum-classical hybrid optimization loop is entered: on the quantum processor, the current parameters are evaluated by running a specially designed local Hilbert-Schmidt test circuit. and Cost function between On a classical computer, a classical optimizer (such as gradient descent) is used to analyze the cost function value and generate a set of updated parameters. and This loop iterates until the cost function value falls below a preset threshold or the maximum number of iterations is reached. The optimized parameters obtained at this point are denoted as... and ,and This is the optimized variable quantum circuit, which is a... A high-quality approximate diagonalized version.

[0075] ③ The optimized variable quantum circuit is embedded as a time evolution submodule into a pre-defined Green's function measurement circuit, and a target quantum circuit for long-term evolution is constructed. The pre-defined Green's function measurement circuit is designed to calculate specific Green's function matrix elements (such as...). The fixed circuit framework is designed to include a mechanism for performing time evolution. The key operation of this invention is to replace this module with... However, in order to calculate arbitrary time intervals... The evolution is not about turning Repeated concatenation Instead, it utilizes the mathematical properties of its diagonalized structure. Specifically, it preserves the circuit structure. Completely unchanged, except for its internal diagonal portion parameter vector Multiply by scaling factor This yields a new diagonal portion. Therefore, it is used to simulate long-term... The quantum module of evolution is .Will By inserting a pre-set Green's function measurement circuit and replacing the original time evolution module, a device for a specific time period is assembled. A target quantum circuit with a fixed circuit depth. For a series of time points... Simply use different scaling factors Generate the corresponding parameters This allows for the rapid configuration of the corresponding target circuit, and all circuits have the same quantum gate depth.

[0076] ④ Run a series of target quantum circuits and obtain the Green's function data in the time domain through quantum measurement, for each selected evolution time. The corresponding target quantum circuit is then compiled and executed on a specific quantum processor. Each circuit needs to be run independently a large number of times, i.e., it undergoes... Sub-sampling measurement. The measurement results of the final state of the circuit (usually the state of the auxiliary qubit) are presented in the form of a probability distribution. The auxiliary qubit is measured statistically. Number of states The corresponding probability estimate can be calculated. According to the mathematical model upon which the preset Green's function measurement circuit is based, this probability value is related to the delayed Green's function. There is a linear transformation relationship between the real and imaginary parts of the . For example, there may be a conversion formula. This formula allows you to calculate the probability value obtained through statistics. Transformed into the Green's function at time points Numerical estimation Iterate through all time points. Then, a set of discrete time-domain data sequences is obtained. This constitutes a function An approximate description.

[0077] ⑤ Performing a frequency domain transformation on the time-domain Green's function data to calculate the local density of states is a sub-step performed on a classical computer. First, necessary preprocessing is performed on the obtained time-series data, such as using... The symmetry is then expanded or smoothed. Next, a positive broadening parameter is selected. (For example This ensures the stability of the numerical integration and reflects the finite energy resolution. Then, for each frequency point of interest... Calculate the approximate value of the Green's function in the complex frequency domain: in These are the weighting coefficients for numerical integration (such as the trapezoidal rule). Finally, based on the definition of local density of states, the final spectral line is calculated and output: function This refers to the local density of states spectrum obtained through the technical solution of this invention, which describes the distribution of electronic states in the energy space of the simulated high-temperature superconducting material system.

[0078] 3) Focus on inspecting the Fermi level (corresponding to) Density of states behavior near the superconducting band gap. A key characteristic of high-temperature superconductors is the superconducting band gap. The existence of . In the density of states spectrum, this means that in The density of states is strongly suppressed or even zero, while... Coherent peaks may appear nearby. Careful analysis of the obtained data is necessary. Curve: Observe whether it is in A noticeable indentation or bandgap appears nearby; measure the width of this bandgap (i.e., the bandgap size). ); Check at the bandgap edge The presence of a steep rise or peak structure at the superconducting transition temperature is a hallmark of the superconducting density of states in Bardeen-Cooper-Schrieffer theory, although this may be modified in strongly correlated systems. Furthermore, the well-known "pseudo-gap" phenomenon in high-temperature superconducting materials occurs at the superconducting transition temperature. This opens a partial bandgap near the Fermi level. During analysis, different chemical potentials can be set... Repeat the above calculations by changing the electron doping concentration and observe the change of the density of states spectrum with doping. This is the key to studying pseudo-gap and superconducting dome.

[0079] 4) The normal state of high-temperature superconducting materials typically exhibits non-Fermi liquid behavior. (In the density of states spectrum...) This may manifest as power-law behavior of the density of states near the Fermi level. Instead of a constant (a characteristic of Fermi liquids), it is necessary to examine the structure of the spectral lines at higher energy scales, such as in... The presence of a nearby Hubbard subband reflects strongly correlated Mott physics. If the model includes interactions that lead to charge density or spin density waves, these ordered states may introduce new features such as splitting or van Hove singularity shifts into the density of states spectrum. By analyzing these features, it is possible to assess which dominant electronic order the system tends to exhibit given model parameters.

[0080] 5) Evaluate the physical properties of the simulated high-temperature superconducting material model. Regarding superconducting properties: If exist A complete bandgap is shown at this point, and... The presence of sharp coherent peaks, combined with the assumption that the system is operating at low temperatures, provides support for the model's superconducting solution. The superconducting bandgap can then be further estimated. With transition integral The ratio is compared with experimental data. Regarding conductivity: if the calculation is performed in a simulated normal state (e.g., by selecting parameters to prevent the system from exhibiting superconductivity), then according to... Whether the density of states is zero is used to determine whether it is a metal or an insulator / semiconductor. Regarding magnetic phases: Although the local density of states primarily reflects charge excitation, specific spectral weighting distributions or bandgap structures can be associated with an antiferromagnetic background. For example, under antiferromagnetic order, the density of states spectrum may exhibit a specific bimodal structure. Ultimately, by systematically changing model parameters (such as...) Doping concentration Quantum simulations can generate phase diagrams that illustrate the evolution from Mott insulators to strange metals and then to superconductors, thus revealing the microscopic electronic structure underlying the complex phase diagrams of high-temperature superconducting materials at the level of quantum computing simulations. The output of this analysis process is a quantitative and qualitative description of the physical properties of high-temperature superconducting materials represented by a specific model, providing data insights from quantum simulations for understanding their superconducting mechanisms.

[0081] (2) When the target strongly correlated system is a transition metal oxide, analyzing its physical properties using the technical solution of this invention requires adapting and executing a complete quantum computing simulation process to address the unique electronic structure complexity of this type of material. Transition metal oxides are a large class of compounds containing transition metal elements (such as Mn, Fe, Co, Ni, Cu) and oxygen elements, and their physical properties are dominated by the strongly correlated electronic behavior of the transition metal d orbitals.

[0082] The physical properties of transition metal oxides refer to the rich and tunable collective phenomena exhibited by these materials. Typical properties include, but are not limited to: metal-insulator transitions caused by strong correlations (such as the Mott transition); various phase transitions accompanied by coupling of lattice, charge, orbital, and spin degrees of freedom, such as ferromagnetism, antiferromagnetism, orbital ordering, charge ordering, and superconductivity under certain specific doping conditions. These properties compete and intertwine, making transition metal oxides an important platform for exhibiting cutting-edge properties such as giant magnetoresistance and multiferroism. Analyzing their physical properties hinges on understanding the ground-state and excited-state behavior of d electrons under various interactions such as crystal field splitting, Hund's coupling, and spin-orbit coupling. The specific process for obtaining this information is as follows: 1) Constructing an effective multiorbital Hamiltonian applicable to specific transition metal oxides. The d orbitals of transition metal ions undergo energy level splitting under a crystal field; therefore, a minimal model typically needs to include multiple d orbital degrees of freedom. For example, for materials with perovskite structures (such as LaMnO3 or LaTiO3), a commonly used model is the multiorbital Hubbard model or its extended forms: In this formula, It is a grid index. It is a track index (such as) , ), It is a spin index. and These are the corresponding creation and annihilation operators. It is a transition integral that depends on the orbit and can be calculated using the Slater-Koster parameters. It is the orbital energy level shift caused by the crystal field. It is the Coulomb interaction parameter between orbits (in situ) and adjacent wait). Describe the Hongde coupling, in the form of ,in It is the Hongde coupling constant. It is a spin operator. It may contain more complex crystal field terms. Depending on the specific oxide being studied, these parameters need to be obtained from first-principles calculations or by fitting experimental data.

[0083] 2) Execute S1 to S5 to obtain the local density of states spectrum under this multi-orbit model, specifically adapting it for the higher complexity of the Hamiltonian. First, based on the above multi-orbit Hamiltonian... Constructing short-time evolution operators Because the number of Hamiltonian terms is greater, the number of Pauli strings mapped to qubits is also greater. This significantly increases the complexity, resulting in a deeper quantum circuit for the constructed short-time evolution operator. Next, the variational fast forward algorithm is used to approximately diagonalize this short-time evolution operator. (Unitial transform part) and diagonal part The circuit template (Ansatz) needs to have sufficient expressive power to handle more complex entanglement and energy spectrum structures, which may require increasing the number of layers. Alternatively, a more general entanglement structure can be adopted. By optimizing the local Hilbert-Schmidt test cost function, the optimized variational quantum circuit is finally obtained. Then, this optimized variable quantum circuit is embedded as a time evolution submodule into a pre-defined Green's function measurement circuit. Here, the Green's function typically needs to be defined for specific orbitals and lattice points, for example... Select the orbit-grid combination of interest (e.g., ... ), by scaling the diagonal parameters To build for different evolution time Target quantum circuits with fixed circuit depth are obtained. These target quantum circuits are operated to acquire time-domain data of the localized Green's function for a specific orbit through quantum measurements. Finally, a Laplace transform is performed on the time-domain data, introducing a positive broadening parameter. The imaginary part of the transformation result is then used for calculation to obtain the transition metal oxides described by the multi-orbital model in specific orbitals. Local density of states on .

[0084] 3) Analyzing the characteristics related to the metal-insulator transition and orbital order in the density of states spectrum reveals that a core issue with transition metal oxides is the interaction between Mott physics and orbital physics. The analysis yields... When analyzing the spectrum, it is necessary to check different orbitals simultaneously. The spectral lines. For the Mott insulator, all orbitals are at the Fermi level. ( A clear bandgap should be observed near the orbital order phase. For orbitally ordered phases, different orbitals... and density of states and They will show significant differences, for example, one of the orbits in One orbital has a higher weight while the other is suppressed, which corresponds to electrons preferentially occupying a certain orbital. Furthermore, it is necessary to observe the position and separation of the Hubbard subbands, the size of the band gap between the upper and lower Hubbard bands, and the interaction strength. Related. If the system is in a bad metal or exotic metal phase, it may be... A non-zero density of states with no obvious quasiparticle peak was observed at the point, and the spectral line shape was anomalous.

[0085] 4) Identify the magnetic correlations and characteristics of other ordered phases reflected in the density of states spectrum. Although the local density of states mainly reflects charge excitations, its details are affected by the spin background. For example, in antiferromagnetic order, due to magnetic Brillouin zone folding, the density of states spectrum may show new peaks or shoulder structures at specific energy positions. If the model explicitly introduces electron-lattice coupling that leads to charge density waves, it may be possible to... Corresponding bandgap characteristics were observed. By comparing the density of states of different orbitals and lattice points (if the nonlocal Green's function was calculated), the arrangement patterns of charge, orbitals, and spins can be inferred. Calculating the density of states in different temperature ranges (by adjusting model parameters or simulating finite temperature effects with initial states) can also track phase transition behavior that varies with temperature, such as the evolution from a low-temperature ordered phase to a high-temperature paramagnetic phase.

[0086] 5) Based on the above analysis, determine the physical state of the simulated specific transition metal oxide model. Regarding conductivity: if all orbitals... And if a finite bandgap exists, then the system is a Mott insulator or a charge-transfer insulator; if at least one orbital is in If a system has a finite density of states and a very small or zero band gap, it may behave as a metal or a half-metal. Regarding orbital order: if the density of states of different orbitals is within a certain range... If the nearby weight distribution is asymmetrical and does not conform to the crystal field expectation, it indicates the presence of orbital polarization or orbital order. Regarding magnetism: Analyze whether the spectral characteristics support the magnetic ground state, based on the magnetically ordered background set by the model; for example, whether specific splitting under antiferromagnetic order appears in the calculated spectrum. Regarding other ordered phases: Check whether there are band gaps or peak structures in the spectral lines that are related to charge order or nematic phase. Finally, by systematically changing key parameters (such as electron filling number)... Coulomb energy With bandwidth ratio Crystal field splitting Hongde Coupling This process can simulate doping or stress effects, thereby drawing complex phase diagrams at the quantum computing level and revealing the microscopic conditions and competitive relationships that give rise to various physical properties of transition metal oxides. The output of this analysis process is an explanation of the microscopic mechanisms underlying the macroscopic physical properties of the transition metal oxide system represented by a specific model.

[0087] The overall objective of this invention is to propose a method that uses the variational fast forward algorithm to accurately measure the time evolution operator in a Green's function measurement circuit, which originally had a depth that increased linearly with time. The data is compressed into a variable quantum circuit module of approximately fixed depth, thereby enabling efficient acquisition of time-domain Green's function data on a quantum processor. With the Green's function in the complex frequency domain And finally, the local density of states of the strongly correlated system is calculated. This provides crucial information for the analysis of material physical properties. The technical solution of this invention differs significantly from existing solutions while also exhibiting synergy. Compared to existing Green's function quantum computing schemes based on Trotter decomposition, its core challenge lies in achieving time evolution. The required quantum circuit depth varies with simulation time. Linear growth easily exceeds the coherence time limit of current noisy quantum processors. This invention creatively transforms the original, random... darkening The circuit is replaced with a parameterized variable quantum circuit structure. Due to the diagonal part The doors are interchangeable, and their powers are... This can be equivalent to a scaling operation on the diagonal parameter. Therefore, long-term evolution Circuit structure and short-time evolution Exactly the same, except the rotation angle of the diagonal gates is scaled up. This design allows the physical circuit depth of the time evolution submodule in the Green's function measurement circuit to be determined for any evolution time. All remain approximately fixed and no longer change. This allows for growth, thus breaking through the time window limitations of traditional schemes. Unlike the original application of the Variational Fast Forward algorithm, which primarily emphasizes long-term dynamic simulations and energy spectrum estimation of quantum systems, this invention differs in that it seamlessly embeds the Variational Fast Forward algorithm as a dedicated submodule into a pre-defined Green's function measurement circuit used to calculate many-body correlation functions. This invention does not merely focus on the evolutionary process itself, but rather on outputting a specific physical observable—namely, the time-domain Green's function. Green's function in the complex frequency domain And the final eigenspectral functions such as the local density of states. —This is the direct objective. This application directly leverages the advantages of the variational fast forward algorithm to address practical problems in condensed matter physics and materials science, analyzing the electronic structure of strongly correlated systems, thus realizing a practical path from algorithmic tools to solving specific physical problems.

[0088] To achieve the above objectives, the technical solution of the present invention includes the following key points: The first key point is to construct and optimize the time evolution submodule based on the variational fast forward algorithm. This submodule optimizes the time evolution submodule by adjusting the target Hamiltonian. Short-time evolution operators This is obtained by variational approximation diagonalization. Specifically, the process involves designing a parameterized variational quantum circuit. ,in This is the part of the unitary transformation. The diagonal part and It is an adjustable parameter; the parameter is adjusted through a classical-quantum hybrid optimization loop. and Minimize measurement and The local Hilbert-Schmidt test cost function for proximity; after optimization, the optimized variable quantum circuit is obtained. For any given time Fast-forward evolution can be achieved by scaling the diagonal parameters: The second key point is to achieve a unified interface between the time evolution submodule and the preset Green's function measurement circuit. The preset Green's function measurement circuit is a system for estimating specific Green's function matrix elements (such as...). The designed quantum circuit framework contains a component that needs to be implemented. The solution of this invention is to replace this module with a fixed-depth evolution module constructed using the variational fast forward algorithm. This replacement ensures that the entire Green's function measurement circuit calculates different times. of At this time, its overall circuit depth remains fixed, thus enabling efficient execution on a quantum processor. The third key point is designing an efficient variational circuit structure tailored to the target Hamiltonian and hardware characteristics. The variational circuit structure refers to the unitary transform part... and diagonal part The specific circuit template directly affects the accuracy and efficiency of variational diagonalization. To achieve sufficiently low compilation errors at lower quantum gate depths, the circuit structure needs to be designed based on the symmetry and entanglement properties of the Hamiltonian in the specific strongly correlated system, as well as the hardware topology of the quantum processor. Typical schemes include: diagonal portion Using Pauli Approximating a diagonal matrix using the truncated summation form of operator strings; unitary transformation part A hierarchical entanglement structure is employed, consisting of alternating stacks of single-qubit rotating gate layers and double-qubit entangled gate layers, to enhance its expressive power and optimize convergence. The fourth key point is directly converting the output of the quantum circuit into the density of states information required for physical analysis. By running the target quantum circuit with an embedded variational fast forward time evolution submodule and performing quantum measurements, a series of time points can be obtained. Green's function data on Subsequently, a numerical Laplace transform was performed on these time-domain data on a classical computer, introducing a positive broadening parameter. To ensure integral convergence and simulate finite energy resolution, the calculation is performed. An approximation. Finally, by taking the imaginary part of the calculation result and performing calculations. This directly yields the local density of states spectrum of the system. This spectral function is a core physical quantity for analyzing the band structure, quasiparticle excitations, and other spectroscopic properties of strongly correlated systems.

[0089] This embodiment uses a small-scale strongly correlated system—the two-grid Hubbard model—as an example to demonstrate the entire process of applying the technical solution of this invention. The Hamiltonian of this model is: in It is a grid. The particle number operator. Select specific parameters. , , The Jordan-Wigner transformation maps fermion operators to the spin operators of qubits, thus transforming the Hamiltonian... Rewritten in standard form as a qubit Hamiltonian: In this form, It is a Pauli string, that is For each qubit index ,have .For example, Represents the action on the second qubit. Gate. Since we consider two lattice points, each containing two degrees of freedom (spin up and spin down), the total number of gates required is... One qubit, specifically: 1) Construct and optimize the time evolution submodule based on the variational fast forward algorithm. First, based on the Hamiltonian... Constructing short-time evolution operators Quantum circuits are typically implemented using first-order Suzuki-Trotter decomposition. Next, variable quantum circuits need to be designed. The variational circuit structure. The unitary transform part is selected. The structure is Substructures with the same layer The process repeats, and several additional single-bit rotating gates are added to increase flexibility. A schematic diagram of its structure is shown below. Figure 2 As shown. The whole The stacked structure is shown in the figure. Figure 3 As shown, including layer And several additional single-bit rotation gates. Truncate to the four-body interaction term (i.e., take...) ,correspond (in the form of...). Then, the parameters are adjusted through a classical-quantum hybrid optimization loop. and To minimize the local Hilbert-Schmidt test cost function To verify the optimization effect, the uniform superposition state (i.e., the quantum state obtained after applying the Hadamard gate to all qubits) was calculated after precise evolution. and variational approximate evolution The fidelity between the subsequent quantum states. The results are as follows: Figure 4 As shown, it can be seen that as the number increases... number of layers , The stronger the expressive ability, the better the optimized With the goal The higher the fidelity, the better the results.

[0090] 2) The optimized variational quantum circuit is embedded into a pre-defined Green's function measurement circuit. The ultimate goal of this invention is to calculate the local state density. As a concrete example, calculation The situation, namely .calculate The frequency domain Green's function needs to be obtained first. The function is a time-domain delayed Green's function. Obtained through Laplace transform. Calculation. The key is to estimate a term called The quantity. This quantity can be estimated using a specific quantum circuit, namely a pre-defined Green's function measurement circuit. In this circuit, the encoded operator... and Part It is a variational circuit structure with degrees of freedom. This circuit ultimately measures an auxiliary qubit, and the measured value is... The probability is denoted as , measured The probability is denoted as The relationship between the two is Time evolution is implemented in this circuit. Part of (i.e.) This is the key aspect that this invention aims to modify. In the prior art, Implemented using Suzuki-Trotter decomposition, its circuit depth varies. Linear growth. In this invention, a variational fast forward method is used instead. A The module is decomposed into three parts through variational approximation diagonalization: After optimization, it can be used. Replace the original circuit For long-term evolution In traditional Suzuki decomposition, series connection is required. indivual The module has a very long circuit. And using... Later, thanks to its mathematical form, it is only necessary to take the diagonal part The parameters are scaled to ,Right now Meanwhile, the physical gate sequence and total length of the circuit remain unchanged, achieving a fixed depth.

[0091] 3) Following the above scheme, optimize the parameters that are scalable. The module is embedded into the Green's function measurement circuit, replacing the original one. Modules are used to form a target quantum circuit with a fixed circuit depth. This is done for a series of time points. By scaling the diagonal parameter Configure the corresponding circuits, run these circuits on the quantum processor, and measure the probability of the auxiliary qubits. Obtaining Green's function data in the time domain Subsequently, these data were processed on a classic computer using stretching parameters. The Laplace transform of the local density of states is calculated, and the imaginary part is computed to obtain the local density of states. The results of this invention were compared with those based on traditional Suzuki-Trotter decomposition methods, and the results are as follows: Figure 5 As shown. The comparison shows that the variational fast forward algorithm is used and number of floors At that time, the obtained density of states spectrum The solution matches the exact solution better and significantly outperforms the Suzuki decomposition method under the same time window constraint. This verifies the advantages of this invention in acquiring data with a longer effective time window, thereby improving frequency domain resolution.

[0092] The technical terms involved in this invention and their corresponding explanations are as follows: 1) Hamiltonian Operators describing the energy and dynamics of a quantum system, whose time evolution is determined by... Decide.

[0093] 2) Fidelity between wave functions: For any two wave functions and The fidelity between them is defined as A larger number indicates higher fidelity. 0 represents that the two states are orthogonal (without overlap), and 1 represents that the two wavefunctions are actually the same. 3) Uniform Superposition State: For a given... A qubit system, initial state is The quantum state obtained by applying a Hardarmad gate to each qubit.

[0094] 4) Time evolution operator: It is the most core and deepest part of quantum simulation.

[0095] 5) Variational Fast Forwarding (VFF): A hybrid quantum-classical algorithm. It first processes algorithms with short time steps. Perform variational approximation diagonalization to obtain After that, it will be a long time The evolution is achieved using fixed-depth lines. And because Composed of commutative gates, it can typically be implemented using parameter scaling. .

[0096] 6) (Real-time) Delayed / Rescheduled Green's function: A commonly used quantity in strongly correlated systems. The definition used in this paper (zero-temperature formulation) can be written as... ,in It is a Fermi-type operator, symbol Represents transpose and complex conjugate, , Representatives opposed to exchanging children, specifically , It is a step function.

[0097] 7) Frequency Domain Green's Function and Density of States (DOS): Through band stretching parameters The Fourier / Laplace integral is obtained Local density of states (in) (For example) .

[0098] 8) Ansatz (variable circuit structure): refers to... and The adjustable parameter circuit structure is the key to the success of VFF.

[0099] 9) LHST Cost Function (Local Hilbert–Schmidt Test): VFF is used to measure... and Measurable cost function for proximity: in, This is the entanglement fidelity term that can be estimated via a short path.

[0100] In the above embodiments, although the steps are numbered S1, S2, etc., they are only specific embodiments given by the present invention. Those skilled in the art can adjust the execution order of S1, S2, etc. according to the actual situation. The scheme after adjusting the order is also within the protection scope of the present invention. It can be understood that in some embodiments, some or all of the above embodiments may be included.

[0101] like Figure 6 As shown, a physical property analysis system 200 for a strongly correlated system according to an embodiment of the present invention includes a diagonalization module 201, an optimization module 202, a target quantum circuit construction module 203, a data acquisition module 204, a frequency domain transformation module 205, and a physical property analysis module 206. The diagonalization module 201 is used to: construct a short-time evolution operator based on the Hamiltonian of the target strongly correlated system, and use the variational fast forward algorithm to approximate the diagonalization of the short-time evolution operator to obtain a parameterized variational quantum circuit containing a unitary transformation part and a diagonal part. The optimization module 202 is used to: optimize the adjustable parameters of the parameterized variable quantum circuit, minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator, and obtain the optimized variable quantum circuit. The target quantum circuit construction module 203 is used to: embed the optimized variable quantum circuit as a time evolution submodule into the preset Green's function measurement circuit, and construct a target quantum circuit with a fixed circuit depth by scaling the parameters of the diagonal part of the optimized variable quantum circuit. The data acquisition module 204 is used to: run the target quantum circuit and acquire the data of the Green's function in the time domain of the strongly correlated system through quantum measurement; Frequency domain transformation module 205 is used to: perform frequency domain transformation on the Green's function data in the time domain, and calculate the density of states of the strongly correlated system; The physical property analysis module 206 is used to analyze the physical properties of strongly correlated systems based on the density of states.

[0102] Optionally, in the above technical solution, the cost function is a local Hilbert-Schmidt test cost function, which is used to measure the degree of proximity between the parameterized variable quantum circuit and the short-time evolution operator. The optimization module 202 is specifically used to: iteratively adjust the adjustable parameters of the parameterized variable quantum circuit through a classical optimizer so that the value of the local Hilbert-Schmidt test cost function is minimized, thereby obtaining the optimized variable quantum circuit.

[0103] Optionally, in the above technical solution, the frequency domain transformation module 205 is specifically used to: perform a Laplace transform or Fourier transform on the data of the Green's function in the time domain, introduce a positive broadening parameter in the transform, take the imaginary part of the result of the integral operation and perform calculation to obtain the density of states of the strongly correlated system.

[0104] Optionally, in the above technical solution, the physical property analysis module 206 is specifically used to: analyze the electronic structure and dynamic behavior of a strongly correlated system within a specific energy range by utilizing the energy distribution information of electronic states reflected by the density of states, and evaluate or determine the electrical conductivity, magnetic phase, or superconducting phase of the strongly correlated system based on the electronic structure and dynamic behavior.

[0105] It should be noted that the beneficial effects of the physical property analysis system 200 for strongly correlated systems provided in the above embodiments are the same as those of the physical property analysis method for strongly correlated systems described above, and will not be repeated here. Furthermore, the system provided in the above embodiments is only illustrated by the division of the above functional modules. In practical applications, the above functions can be assigned to different functional modules as needed, that is, the system can be divided into different functional modules according to the actual situation to complete all or part of the functions described above. In addition, the system and method embodiments provided in the above embodiments belong to the same concept, and their specific implementation process is detailed in the method embodiments, and will not be repeated here.

[0106] An electronic device according to an embodiment of the present invention includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the physical property analysis method of any of the above-mentioned strongly correlated systems.

[0107] An embodiment of the present invention provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the physical property analysis method for any of the aforementioned strongly correlated systems.

[0108] The above description is merely a preferred embodiment of the present invention and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of disclosure in this invention is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-disclosed concept. For example, technical solutions formed by substituting the above features with (but not limited to) technical features with similar functions disclosed in this invention.

[0109] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. A method for analyzing the physical properties of a strongly correlated system, characterized in that, include: Based on the Hamiltonian of the target strongly correlated system, a short-time evolution operator is constructed, and the variational fast forward algorithm is used to approximate the diagonalization of the short-time evolution operator to obtain a parameterized variational quantum circuit containing a unitary transformation part and a diagonal part. The adjustable parameters of the parameterized variable quantum circuit are optimized to minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator, thus obtaining the optimized variable quantum circuit. The optimized variable quantum circuit is used as a time evolution submodule and embedded into a preset Green's function measurement circuit. By scaling the parameters of the diagonal part of the optimized variable quantum circuit, a target quantum circuit with a fixed circuit depth is constructed. Run the target quantum circuit and obtain the time domain data of the Green's function of the strongly correlated system through quantum measurement; The density of states of the strongly correlated system is calculated by performing a frequency domain transformation on the time domain data of the Green's function. Based on the density of states, the physical properties of the strongly correlated system are analyzed.

2. The method for analyzing the physical properties of a strongly correlated system according to claim 1, characterized in that, The cost function is a local Hilbert-Schmidt test cost function, which is used to measure the degree of similarity between the parameterized variable quantum circuit and the short-time evolution operator. Optimizing the adjustable parameters of the parameterized variable quantum circuit to minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator yields the optimized variable quantum circuit, including: The adjustable parameters of the parameterized variable quantum circuit are iteratively adjusted using a classical optimizer to minimize the value of the local Hilbert-Schmidt test cost function, thus obtaining the optimized variable quantum circuit.

3. The method for analyzing the physical properties of a strongly correlated system according to claim 2, characterized in that, The density of states of the strongly correlated system is calculated by performing a frequency domain transformation on the time domain data of the Green's function, including: The Green's function is subjected to a Laplace transform or a Fourier transform in the time domain. A positive broadening parameter is introduced into the transform. The imaginary part of the integral is taken and calculated to obtain the density of states of the strongly correlated system.

4. The method for analyzing the physical properties of a strongly correlated system according to claim 3, characterized in that, Based on the density of states, the physical properties of the strongly correlated system are analyzed, including: By utilizing the energy distribution information of electronic states reflected by the density of states, the electronic structure and dynamic behavior of the strongly correlated system within a specific energy range are analyzed, and the conductivity, magnetic phase, or superconducting phase of the strongly correlated system are evaluated or determined based on the electronic structure and dynamic behavior.

5. A physical property analysis system for strongly correlated systems, characterized in that, It includes a diagonalization module, an optimization module, a target quantum circuit construction module, a data acquisition module, a frequency domain transformation module, and a physical property analysis module; The construction of the diagonalization module is used to: construct a short-time evolution operator based on the Hamiltonian of the target strongly correlated system, and use the variational fast forward algorithm to approximate the diagonalization of the short-time evolution operator to obtain a parameterized variational quantum circuit containing a unitary transformation part and a diagonal part. The optimization module is used to: optimize the adjustable parameters of the parameterized variable quantum circuit, minimize the cost function between the parameterized variable quantum circuit and the short-time evolution operator, and obtain the optimized variable quantum circuit. The target quantum circuit construction module is used to: embed the optimized variable quantum circuit as a time evolution submodule into a preset Green's function measurement circuit, and construct a target quantum circuit with a fixed circuit depth by scaling the parameters of the diagonal part of the optimized variable quantum circuit. The data acquisition module is used to: run the target quantum circuit and acquire the Green's function of the strongly correlated system in the time domain through quantum measurement; The frequency domain transformation module is used to: perform frequency domain transformation on the Green's function data in the time domain, and calculate the state density of the strongly correlated system; The physical property analysis module is used to analyze the physical properties of the strongly correlated system based on the density of states.

6. The physical property analysis system for a strongly correlated system according to claim 5, characterized in that, The cost function is a local Hilbert-Schmidt test cost function, which is used to measure the degree of similarity between the parameterized variable quantum circuit and the short-time evolution operator. The optimization module is specifically used to: iteratively adjust the adjustable parameters of the parameterized variable quantum circuit using a classical optimizer, so as to minimize the value of the local Hilbert-Schmidt test cost function, thereby obtaining the optimized variable quantum circuit.

7. The physical property analysis system for a strongly correlated system according to claim 6, characterized in that, The frequency domain transformation module is specifically used to: perform a Laplace transform or Fourier transform on the time domain data of the Green's function, introduce a positive broadening parameter in the transform, take the imaginary part of the result of the integral operation and perform calculations to obtain the density of states of the strongly correlated system.

8. The physical property analysis system for a strongly correlated system according to claim 7, characterized in that, The physical property analysis module is specifically used to: analyze the electronic structure and dynamic behavior of the strongly correlated system within a specific energy range by utilizing the energy distribution information of the electronic states reflected by the density of states, and evaluate or determine the electrical conductivity, magnetic phase, or superconducting phase of the strongly correlated system based on the electronic structure and dynamic behavior.

9. An electronic device, characterized in that, It includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement a method for analyzing the physical properties of a strongly correlated system as described in any one of claims 1 to 4.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program, which, when executed by a processor, implements the physical property analysis method for a strongly correlated system as described in any one of claims 1 to 4.

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