A method for modeling and optimizing compound noise in an NISQ device quantum circuit

Abstract

This invention belongs to the field of quantum computing technology and relates to a method for modeling and optimizing composite noise in the quantum circuits of NISQ devices. Based on the Lindebrade master equation, a composite noise model integrating relaxation noise and phase noise is constructed, and the coupling effect of multi-source noise is simulated using Kraus operators. An improved zero-noise extrapolation technique is proposed, employing a zero-noise neural network extrapolation method. A multilayer perceptron is used to replace the analytical extrapolation function, transforming error correction from a fitting problem into a data-driven prediction problem. Using the one-dimensional transverse field Ising model as a benchmark, a ground state preparation and dynamic evolution circuit is constructed in a quantum cloud computing platform and Cqlib environment. Multi-noise level data is generated using a time scaling strategy to train the neural network. This invention reduces the average absolute error by 92.2% across the entire parameter region, including ferromagnetic phases, paramagnetic phases, and quantum critical points, demonstrating superior accuracy and stability compared to traditional extrapolation methods.

Description

Technical Field

[0001] This invention belongs to the field of quantum computing technology, specifically relating to a method for modeling and optimizing composite noise in the quantum circuit of NISQ devices. Background Technology

[0002] Quantum computing, with its properties such as quantum superposition and entanglement, has shown great potential in solving complex problems. However, quantum hardware in the NISQ era faces severe noise interference. Quantum bit decoherence (such as relaxation and phase dissipation) and quantum gate operation errors can lead to deviations in calculation results, hindering the practical application of quantum computing. Existing noise suppression techniques suffer from two major bottlenecks: First, noise modeling is simplified. Existing models often treat different noise sources as independent channels, making it difficult to characterize the coupling effect and nonlinear characteristics of multi-source noise in real devices, resulting in limited effectiveness of noise suppression strategies. Second, error mitigation methods have poor adaptability. Traditional zero-noise extrapolation (ZNE) techniques rely on the prior selection of analytical extrapolation functions such as linear and polynomial functions, which cannot dynamically adapt to hardware heterogeneity and time-varying noise, leading to a significant decrease in accuracy in complex regions such as quantum critical points.

[0003] Therefore, there is an urgent need to develop an efficient noise suppression scheme that can accurately model composite noise and dynamically adapt to hardware characteristics in order to improve the quantum simulation fidelity of NISQ devices. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for modeling and optimizing composite noise in quantum circuits of NISQ devices.

[0005] To achieve the objectives of this invention, the following technical solutions are adopted.

[0006] S1. In the quantum computing platform and Cqlib programming framework, a quantum circuit consisting of a ground state preparation circuit and a dynamic evolution circuit is constructed using the one-dimensional transverse field Ising model. A scalable composite noise model that comprehensively considers the coupling effect of decoherence and gate operation noise sources is inserted into the quantum circuit. The scalable composite noise model is based on the Lindeblad master equation and is constructed by integrating relaxation noise and phase noise through the Kraus operator. The generators of its dynamic evolution are additive.

[0007] S2. Using the time scaling extrapolation method, the effective execution time of the quantum gate is adjusted by the noise scaling factor to generate the expected value of the quantum circuit under different noise intensities, i.e., the expected value of multiple noise levels.

[0008] S3. Using the expected value of the multiple noise levels as input, a zero-noise extrapolation neural network is used for function fitting to output the predicted value under the zero-noise limit, thereby completing noise suppression; wherein: the zero-noise extrapolation neural network is formed by training the model parameters of a multilayer perceptron using the expected value of the multiple noise levels as training samples, the analytical solution of the one-dimensional transverse field Ising model as the supervision signal, and minimizing the mean square error as the objective; the multilayer perceptron includes an input layer, six hidden layers, and an output layer, with the number of neurons in the hidden layers decreasing in a descending order.

[0009] Furthermore, the quantum circuit is used to simulate the one-dimensional transverse field Ising model, and its Hamiltonian is:

[0010] ,

[0011] Quantum circuits achieve ground state preparation and dynamic evolution through Jordan-Wigner transform, Fourier transform, and Bogoliubov transform, and the circuit depth does not change with evolution time.

[0012] Furthermore, the relaxation noise The corresponding Kraus operator set is:

[0013] ,

[0014] The phase noise The corresponding Kraus operator set is:

[0015] ,

[0016] in, , , This represents the original execution time of the quantum gate.

[0017] Furthermore, the additivity is expressed as follows:

[0018] ,

[0019] in, For relaxation noise dissipation operators, For phase noise dissipation operators, For the system Hamiltonian, This is the system density matrix.

[0020] Furthermore, the time scaling strategy maintains the quantum gate pulse area unchanged and scales it proportionally. Stretch execution time And compress the amplitude ,satisfy: , .

[0021] Furthermore, the multilayer perceptron employs the ReLU activation function, introduces Dropout layer regularization, and minimizes the loss function using the Adam optimization algorithm.

[0022] ,

[0023] in, For network parameter set, This is the analytical solution for the Ising model. This represents the number of training samples.

[0024] A noise suppression system for quantum circuitry in a NISQ device, comprising:

[0025] The quantum circuit construction module, within the quantum computing platform and Cqlib programming framework, utilizes the one-dimensional transverse field Ising model to construct quantum circuits consisting of ground state preparation circuits and dynamic evolution circuits. A scalable composite noise model, comprehensively considering the coupling effects of decoherence and gate operation noise sources, is inserted into the quantum circuits. Specifically, the scalable composite noise model is constructed using the Lindeblad master equation, integrating relaxation noise and phase noise through Kraus operators, and its dynamic evolution generators possess additive properties.

[0026] The multi-noise data generation module uses a time scaling extrapolation method to generate the expected values ​​of quantum circuit observation under different noise intensities, i.e., multi-noise level expected values, by adjusting the effective execution time of the quantum gate through a noise scaling factor.

[0027] The noise suppression execution module takes the expected values ​​of the multiple noise levels as input, performs function fitting through a zero-noise extrapolation neural network, and outputs the predicted value under the zero-noise limit to complete noise suppression. The zero-noise extrapolation neural network is trained by a neural network training module. This module uses the expected values ​​of the multiple noise levels as training samples, the analytical solution of the one-dimensional transverse field Ising model as the supervision signal, and minimizes the mean square error as the objective. It trains the model parameters of a multilayer perceptron to form the zero-noise extrapolation neural network. The multilayer perceptron includes an input layer, six hidden layers, and an output layer, with the number of neurons in the hidden layers decreasing in a descending order.

[0028] Furthermore, in a multi-qubit system, the coupling effect is achieved by extending the single-bit noise Kraus operator to the global Hilbert space through tensor product.

[0029] A computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps of a method for modeling and optimizing composite noise in a quantum circuit of an NISQ device.

[0030] Compared with existing technologies, it has the following beneficial effects:

[0031] (1) More accurate noise modeling: integration of composite noise models The coupling effect of noise is simulated physically realistically using Kraus operators, providing a reliable basis for noise suppression;

[0032] (2) Higher suppression accuracy: The NN-ZNE method utilizes the nonlinear fitting capability of the multilayer perceptron to reduce the average absolute error by 92.2% in the full parameter region, including ferromagnetic phase, paramagnetic phase and quantum critical point, which is significantly better than the traditional extrapolation method;

[0033] (3) More adaptable: Data-driven prediction models do not require prior function assumptions and can dynamically adapt to the noise characteristics of different hardware, solving the problems of noise heterogeneity and time drift.

[0034] (4) Controllable resource overhead: Based on the time scaling strategy, no additional qubits are required, adapting to the hardware limitations of NISQ devices, and can be directly integrated into the Tianyan platform and Cqlib framework. Attached Figure Description

[0035] Figure 1 Flowchart for the NN-ZNE method optimization of the quantum circuit of the composite noise model;

[0036] Figure 2 The circuit diagram for generating the ground state of the Ising model of the "Tianyan" quantum computing platform is shown below.

[0037] Figure 3 This is a graph showing the dependence of transverse magnetization on transverse field strength λ in the ground state of the four-bit Ising model described in this invention.

[0038] Figure 4 A comparison graph showing the noiseless evolution theoretical solution (solid line) and simulation results (dotted line) under different transverse field intensities λ;

[0039] Figure 5 Comparison of time evolution curves of the original data with noise, various ZNE fitting methods, NN-ZNE optimization method and theoretical analytical solution under transverse field intensities (a) λ=0.5; (b) λ=0.9; (c) λ=1.8;

[0040] Figure 6 A quantitative comparison of the error improvement effects of different ZNE extrapolation fitting methods;

[0041] Figure 7 A comparison of the time evolution curves of the NN-ZNE method and the cubic function fitting method under different λ values;

[0042] Figure 8 The graph shows the error improvement factor achieved by the NN-ZNE method as a function of the transverse field intensity λ. Detailed Implementation

[0043] The present invention will be further described in conjunction with the embodiments and accompanying drawings.

[0044] To realize the potential of the hybrid quantum-classical framework, this invention will address two key aspects. First, to more accurately evaluate and guide noise suppression strategies, this invention will construct a composite noise model that reflects real physical processes. This model will comprehensively consider the coupling effects of multiple noise sources, such as decoherence and gate operation errors, aiming to provide a more solid theoretical foundation for understanding and simulating the noise behavior of NISQ devices. Second, regarding noise suppression methods, this invention will propose an innovative scheme combining Zero-Noise Extrapolation (ZNE) with Multi-Layer Perception (MLP) in machine learning. This invention aims to utilize the powerful nonlinear learning capabilities of MLP to analyze noisy data during the ZNE process, thereby learning and predicting the noise patterns of specific hardware to achieve more accurate and adaptive error suppression than traditional methods. Through the organic combination of these two parts, this invention aims to provide a closed-loop solution from accurate modeling to adaptive suppression for achieving high-fidelity quantum many-body fabrication in the NISQ era.

[0045] The specific path is as follows: Figure 1 As shown, a complete and physically meaningful experimental circuit was first constructed based on the Tianyan quantum cloud computing platform and the Cqlib environment. This invention utilizes the Tianyan quantum cloud computing platform to construct the ground state generation quantum circuit of the four-qubit Ising model and the dynamic evolution quantum circuit of the Ising model.

[0046] Based on the constructed quantum circuit, this invention first introduces scalable composite noise at the code level, such as... , This paper considers the coupling effects of multiple noise sources, such as decoherence and relaxation, to comprehensively address the combined noise. To optimize the results for noisy circuits, this invention introduces a Neural Network Assisted Zero-Noise Extrapolation (NN-ZNE) method, which combines zero-noise extrapolation with a multilayer perceptron. By using a neural network to replace function fitting in the general ZNE method, more efficient and accurate results can be achieved. As shown in Table 1, the experimental simulation results demonstrate that NN-ZNE significantly improves the accuracy of the results. Compared with the original data generated from noisy circuits, the average improvement is as high as 92.2%.

[0047] 1. Construction of the composite noise model

[0048] In the grand vision of quantum computing, qubits, as the fundamental carriers of information, directly determine the upper limit of computing power based on their ability to retain information. However, qubits are highly susceptible to interference from their surroundings, leading to decoherence. This unavoidable noise interaction is one of the core bottlenecks limiting the performance of quantum computers and hindering their practical application.

[0049] To accurately understand, simulate, and ultimately suppress these noises, it is crucial to establish a noise model that accurately reflects the characteristics of physical devices. Based on the aforementioned Linblad master equation, we construct the most fundamental and ubiquitous mechanisms among numerous noise sources—relaxation noise and phase noise. We then build a composite noise model based on these two noise mechanisms, forming the main framework for describing the decoherence process of qubits.

[0050] (1) Relaxation noise

[0051] Relaxation noise essentially originates from the irreversible transfer of energy from the system to its external environment. In systems such as superconducting qubits, this process typically manifests as occurring in an excited state. The qubit decays to its ground state through channels such as spontaneous emission. That is, the so-called This process not only directly alters the energy distribution of the system but also disrupts the coherence of the quantum states.

[0052] At the level of mathematical modeling, relaxation noise is typically reduced by Pauli operators. Characterized by its role in the Lindbrard equation:

[0053] ,

[0054] In this equation, the relaxation rate The physical meaning of this term is that the probability of the system decaying from the excited state evolves over time, and this process is considered a memoryless Markov process. If expressed using Kraus operators, the relaxation noise corresponds to the amplitude damping channel, and its Kraus operator set is:

[0055] , ,

[0056] in, This represents the time interval. The probability of decay occurring within the circuit. This formulation provides a direct tool for numerically simulating relaxation noise at the quantum circuit level.

[0057] (2) Phase noise

[0058] Phase noise, also known as dephase noise, is another prevalent decoherence mechanism. It primarily originates from low-frequency or quasi-static random fluctuations in the environment. These fluctuations cause random drifts in the system's Hamiltonian eigenvalues ​​(i.e., energy levels), leading to random accumulation of the relative phases between different components in the quantum superposition state. Unlike relaxation noise, pure phase noise does not involve energy exchange between the system and its environment, but it still severely disrupts quantum coherence. This process is typically caused by… Time is used to characterize it.

[0059] Within the framework of the Lindebrade master equation, pure phase noise can be expressed by the Pauli operator. Characterized by the supercomputing terms of their effects:

[0060] ,

[0061] Dephase rate here Total out-of-phase time and relaxation time Related, their relationship is ,and Its effect manifests as the off-diagonal elements of the density matrix... The process decays exponentially. In the Kraus operator representation, this process corresponds to a phase-damped channel, whose Kraus operator set is:

[0062] , ,

[0063] in: Is with The probability is time-dependent. The composite noise model constructed by combining these two noise channels can more realistically reflect the main decoherence process experienced by qubits in NISQ devices, thus providing a more reliable simulation basis for subsequent research on noise suppression strategies.

[0064] (3) Construction of the composite noise model

[0065] Accurate simulation of noise behavior in real quantum devices requires models capable of describing the concurrent effects of multiple decoherence mechanisms. Therefore, the noise model used in this invention is not based on a single noise process, but rather incorporates relaxation (…). ) and phase ( The noise effects are systematically composited. This composite structure has a consistent physical core under different theoretical formulations. Within the framework of the Lindblad master equation, the overall dynamic evolution of the system is a linear superposition of coherent evolution and the dissipation terms of each independent noise process, and its generators are additive:

[0066] ,

[0067] in, and These correspond to the relaxation and phase noise dissipation operators, respectively. Equivalently, from the perspective of quantum channels, which are closer to quantum circuit implementations, the composite noise effect manifests as a mapping combination of multiple independent noise channels. Within a discrete time step, the system's state transformation... This can be understood as the sequential action of the coherent evolution channel and each decoherent channel.

[0068] A core advantage of this model-building method lies in its inherent modularity and the resulting high scalability. This framework allows researchers to progressively increase the complexity and realism of the model by adding new physics terms. For example, systematic coherent control errors, such as pulse over-spinning or under-spinning caused by inaccurate microwave field calibration, can be addressed by introducing perturbation terms into the Hamiltonian. To describe this, for scalable multi-qubit systems, crosstalk between qubits introduces spatially correlated noise. This process cannot be fully characterized at the single-qubit level and must be described using methods such as... The form of multi-body dissipation operators. This flexible modeling strategy provides a powerful numerical simulation platform for systematically evaluating and optimizing the performance of error correction schemes such as ZNE.

[0069] 2. Zero-noise extrapolation method (ZNE)

[0070] Zero-noise extrapolation (ZNE) is a highly practical quantum error mitigation technique in the NISQ era. Its core idea is not to directly modify the quantum gate operations themselves, but rather, while maintaining the logical function of the quantum circuit, to artificially and controllably amplify the noise intensity in the system, measure the expected value of the output observable at different noise levels, and finally extrapolate the result to the ideal limit where the noise intensity is zero. Compared to resource-intensive quantum error-correcting codes, ZNE does not require the introduction of additional qubits, making it a lightweight and efficient error mitigation scheme under current hardware conditions. The workflow of the ZNE method is as follows: Figure 1 As shown, the theoretical basis of the ZNE method lies in a key assumption: when the noise intensity... When small, the expected value of the observable Taylor can expand this into a discussion about Analytical functions:

[0071] ,

[0072] in: This is the ideal noise-free value we are looking for. The actual operation of ZNE involves measuring a series of known noise levels. Expected value Then, using these data points, a suitable function model (such as a polynomial) is fitted, and finally, the extrapolation is performed. intercept at time In order to achieve the noise multiplier experimentally Controllable amplification typically employs the following two mainstream methods:

[0073] Gate Folding is a technique that amplifies noise at the digital (gate level). Its principle is to manipulate a noisy door. Then, immediately follow by inserting its ideal inverse operation. Compared with the original operation The combination, namely In an ideal situation, This constitutes a single operation and does not change the logical function of the circuit. However, in the physical implementation, each gate operation (regardless of whether it is a single operation or a single operation) is a single operation. still Each of these steps introduces a base noise level. Therefore, repeated insertions... Second-rate Yes, the noise introduced by this gate can be approximately amplified without changing its logical function. This method is particularly sensitive to gate operation fidelity errors and is easy to implement at the compiler level.

[0074] Time scaling is another strategy for amplifying noise at the physical (pulse level), and it is particularly suitable for quantum hardware dominated by decoherence, such as superconducting qubits. The basic idea is to maintain the performance of a unitary operation (e.g., single-qubit rotation) while simultaneously scaling the qubit. Under the premise that the total pulse area remains unchanged, its duration is... proportionally Stretch, while increasing its amplitude Accordingly Proportional compression:

[0075]

[0076] In this way, ideal logical operations are maintained, but due to the extended operation time, the time the qubit is exposed to environmental noise (such as relaxation noise and phase noise) is also synchronously amplified, thereby achieving controllable enhancement of noise intensity. Compared with gate folding, time scaling directly modulates time-dependent decoherent noise, avoiding the introduction of additional logic gates, and in some cases can achieve more linear and precise noise level control. Subsequent experiments of this invention will mainly use the time scaling method to generate evolution data under different noise levels.

[0077] 3. Neural Network Zero-Noise Extrapolation Method (NN-ZNE)

[0078] In traditional zero-noise extrapolation processes, the accuracy of the extrapolation results highly depends on prior assumptions about the functional relationship between noise and the expected value. Simple fitting models, such as linear or low-order polynomials, while performing reasonably well under certain idealized noise conditions, often fail to capture the complex noise behavior of real NISQ devices. In actual hardware, relaxation, dephase, control errors, and the potential nonlinear coupling between them collectively lead to a complex and nontrivial dependency between the measurement results and the noise amplification factor. To overcome this limitation of traditional extrapolation methods, this invention proposes integrating the powerful machine learning tool, the multilayer perceptron (MLP), into the ZNE framework, combining them into a neural network zero-point extrapolation method. Its workflow diagram is shown below. Figure 1 As shown. Our aim is to leverage the superior nonlinear function approximation capabilities of MLPs to learn the underlying noise patterns directly from noisy measurement data through a data-driven approach, thereby achieving more accurate predictions of the zero-noise limit.

[0079] Within the framework of the neural network zero-point extrapolation method of this invention, the input layer of the MLP is designed to receive a set of signals at different noise levels. Expected value obtained from the measurement These data points are processed through several hidden layers connected by non-linear activation functions (such as ReLU), and finally the output layer provides a single predicted value. Its forward propagation process can be formally represented as:

[0080] ,

[0081] ,

[0082] in: It is the input vector. and These are the trainable parameters of the network, and This is a non-linear activation function. The training objective of the model is to optimize the network parameters by minimizing the mean square error (MSE) between the predicted values ​​and the true noise-free values ​​(obtained through numerical simulation or theoretical calculation). :

[0083] ,

[0084] By training on a large amount of simulated data, the MLP can autonomously learn and internalize complex noise-result mappings without any artificial assumptions about the analytical form of the noise model. During the inference phase, when faced with new experimental data, the trained MLP model only needs to input measurements at a few different noise levels to directly output an optimized zero-noise prediction. This approach transforms the heuristically chosen "fitting" step in traditional ZNE into a systematic, learnable "prediction" task. Especially in scenarios with complex hardware noise characteristics or high sampling costs, this method is expected to exhibit higher accuracy and robustness compared to traditional polynomial extrapolation.

[0085] To effectively verify the theoretical model and error suppression strategy proposed in this invention numerically, we designed and implemented a comprehensive quantum simulation and analysis framework. This chapter will elaborate on the core components of this framework, covering the implementation of a high-fidelity noise model, the decomposition strategy of key quantum gates, the construction logic of quantum circuits, and the specific design of the machine learning optimization module.

[0086] 4. Implementation of the composite noise model

[0087] To realistically evaluate the performance of various error suppression strategies in near-real-world hardware environments, we first constructed a high-fidelity noise simulation framework. The core of this framework is a density matrix-based quantum simulator, a crucial choice for accurately capturing the mixed-state evolution caused by environmental decoherence effects. We focus on the two most prevalent noise sources in quantum hardware—relaxation noise (… (noise) and phase noise ( Noise) — was modeled.

[0088] In our implementation, these noisy processes are all expressed using the mathematical form of Kraus operators. Relaxation noise, whose corresponding amplitude damping channel is defined by the following Kraus operator:

[0089] , ,

[0090] Among them, the attenuation probability Duration of AND gate operation and system Time is directly relevant. Similarly, for The phase-damped channel has the following Kraus operator:

[0091] , ,

[0092] Here In a multi-qubit system, the action on a single qubit... The noise channel, whose Kraus operator is extended to the global Hilbert space via tensor product, i.e. .

[0093] In the simulation process, the complete evolution of each quantum gate operation is broken down into two steps: first, the ideal unitary evolution. Next, the corresponding noise channels are sequentially applied to all qubits participating in the gate operation. The entire process is precisely described by Algorithm 1.

[0094] Algorithm 1 is a density matrix simulation algorithm for noisy evolution:

[0095]

[0096] 5. Key Quantum Gate Decomposition Module

[0097] The key to effectively mapping the theoretical solution process of the Ising model to quantum circuits lies in the ability to precisely decompose the abstract unitary transformations (such as Fourier transforms and Bogoryubov transforms) involved in the theory into a sequence of basic quantum gates that can be executed on the target quantum hardware. To this end, we have specifically designed and implemented a quantum gate decomposition module.

[0098] The core function of this module is to provide accurate decomposition schemes based on a universal quantum gate set for various composite gates required in the Ising model simulation. We employ a recursive decomposition strategy to ensure that each composite gate, including the controlled Hadamard (CH), fermion commutation gate (FSWAP), controlled RX gate (CRX), Bogoryubov transform gate, and Fourier gate, can be accurately represented as an ordered combination of a series of standard single-qubit and two-qubit gates (e.g., CZ gates). The mathematical accuracy of these decomposition schemes has been rigorously verified, forming the fundamental premise for ensuring the physical fidelity of the entire simulation results.

[0099] 6. Logic for constructing time evolution circuits

[0100] The time evolution circuit design in this invention aims to accurately simulate the four-bit Ising Hamiltonian:

[0101] ,

[0102] The dominant quantum dynamics process. Our construction logic does not adopt the approximate Trotterization method, but fully utilizes the analytical solvability of the model, constructing evolution operators through precise diagonalization, thereby avoiding the introduction of Trotter error in principle.

[0103] As mentioned earlier, the complete evolution operator It can be broken down into a three-part structure: .

[0104] Our circuit construction is a direct "translation" of this process:

[0105] Phase 1: Diagonalization transformation. The circuit first applies... Transformation, this operation converts the system from computational basis vectors to eigenvalue basis vectors of the Hamiltonian.

[0106] Second stage: Diagonalization evolution. Under the eigenbase, time evolution simplifies to applying a phase rotation independently to each eigenstate, the rotation angle of which is determined by the corresponding eigenenergy and evolution time. A joint decision.

[0107] Third stage: Inverse diagonalization transformation. Finally, the circuit is applied... The transformation converts the system from the intrinsic basis back to the computational basis for subsequent measurements.

[0108] The key advantage of this design is that it is effective regardless of evolutionary time. Regardless of changes, the structure and depth of the entire quantum circuit remain strictly constant. This property is crucial for the subsequent implementation of time-scaling zero-noise extrapolation methods. Algorithm 2 illustrates this construction process in detail.

[0109] Algorithm 2 is an algorithm for constructing time-evolutionary circuits:

[0110]

[0111] 7. Machine Learning-Based Optimization Framework

[0112] To overcome the inherent limitations of analytical extrapolation function selection in traditional ZNE methods, we introduce a machine learning-based optimization framework. At the core of this framework is a deep neural network specifically designed for ZNE optimization tasks, aiming to learn a complex nonlinear mapping from measurements at a range of different noise scales to an ideal noise-free result.

[0113] The network architecture employs a standard feedforward neural network, consisting of one input layer, six hidden layers, and one output layer. The number of neurons in the hidden layers is designed to decrease progressively ([256, 128, 64, 32, 16]) to achieve layer-by-layer feature extraction and compression. We choose ReLU as the activation function and introduce a Dropout layer for regularization to enhance the model's generalization ability. During training, we use the Adam optimization algorithm with mean squared error as the loss function, combined with an early stopping mechanism to effectively prevent overfitting. Crucially, the network training process uses high-precision theoretical values ​​obtained from the analytical solution of the Ising model.

[0114] As a supervisory signal (i.e., the target value), this provides a solid guarantee for the model to learn the correct physical laws.

[0115] 8. Implementation of zero-noise extrapolation for time scaling

[0116] In our zero-noise extrapolation implementation, we employ a time-scaling strategy to artificially amplify the noise intensity, contrasting with traditional gate-folding methods. Gate-folding accumulates noise by repeatedly executing quantum gates, but this causes the circuit depth to increase linearly with the noise scaling factor, introducing additional computational complexity and coherence errors. The time-scaling method cleverly circumvents this problem by scaling the effective noise intensity solely by adjusting the physical execution time of the gate operations, while maintaining the quantum circuit topology completely unchanged.

[0117] Its physical principle is rooted in the exponential decay characteristic of quantum decoherence processes. This is achieved by introducing a dimensionless noise scaling factor. We define the effective gate action time as This causes the attenuation probability in the noise channel to become:

[0118] ,

[0119] ,

[0120] This method not only ensures the continuous adjustability of noise intensity, but also decouples the circuit depth from the noise amplification process.

[0121] In obtaining a series of different noise scales After obtaining the measurement data, our framework implements six traditional analytical extrapolation methods in parallel (linear, quadratic polynomial, cubic polynomial, exponential, Richardson, and rational function extrapolation), as well as our proposed machine learning-based augmentation method. The latter completely bypasses the prior assumptions about the functional form and directly predicts the ideal zero-noise result from the noisy measurement vector through a trained neural network.

[0122] 9. Comprehensive Simulation and Analysis Process

[0123] To ensure the systematic nature and reproducibility of the research, we established a complete integrated simulation and analysis workflow, which coordinates the collaborative work of all the aforementioned components. The system architecture adopts a modular design, ensuring loose coupling and high cohesion between various functional modules (such as ground state calculation, dynamic evolution, error analysis, etc.), facilitating future functional expansion and performance optimization.

[0124] The core of the entire process is the time evolution simulation module, which can simulate the time evolution at a given coupling strength. and evolution time A systematic scan is performed in the two-dimensional parameter space. For each parameter point... The system will perform noise-free benchmark calculations in parallel, as well as calculations using various error mitigation strategies in noisy environments.

[0125] Subsequently, the error analysis subsystem performs multi-dimensional performance evaluations of the results, calculating metrics including, but not limited to, mean absolute error (MAE), root mean square error (RMSE), and percentage improvement. All simulation data is stored in the efficient NumPy compression format, while statistical analysis results are exported in the common CSV format for subsequent processing. Finally, the visualization subsystem generates a series of charts, from single-parameter curves to multi-dimensional heatmaps, providing intuitive support for the results analysis.

[0126] 10. Results Analysis

[0127] (1) Circuit verification in a noise-free environment

[0128] Before introducing a noise model and evaluating error suppression strategies, the primary task is to ensure the correctness of the designed quantum circuit in an ideal (i.e., noise-free) environment. To this end, we conducted simulation experiments based on the constructed four-qubit Ising model ground state generation circuit and dynamic evolution circuit, both locally and on the full-amplitude simulator of the Tianyan quantum computing cloud platform.

[0129] We first tested the effectiveness of the ground state generation circuit, as shown in the circuit diagram. Figure 2 As shown, the ground state was prepared under different transverse field intensities λ, and the average transverse magnetization of the system was calculated. We obtained the following: Figure 3 The results are shown. From Figure 3It can be clearly observed that at the critical point where the transverse field strength λ=1, the slope of the system's average magnetization curve undergoes a sudden change. This phenomenon precisely corresponds to the theoretically predicted quantum phase transition from the ferromagnetic phase to the paramagnetic phase. The simulation results are in high agreement with the known theoretical solutions, which fully confirms the correctness and effectiveness of our designed ground state generation circuit, demonstrating that it can accurately prepare the ground state under different parameters and successfully capture the core physical characteristics of the system.

[0130] Building upon this, we further validated the dynamic evolution circuit. The experiment selected three typical transverse field parameters (λ=0.5, 0.9, 1.8) representing the ferromagnetic phase, the vicinity of the critical point, and the paramagnetic phase, respectively, using their corresponding ground states as the initial states, and then performed time evolution using the designed circuit. Figure 4 The simulation results and theoretical analytical solutions for the evolution of the system's average magnetization over time are presented. As shown in the figure, for all tested λ values, the numerical results (dotted lines) obtained using our quantum circuit on the Cqlib simulator almost completely coincide with the theoretical analytical solutions (solid lines). This result not only reaffirms the accuracy of the ground state preparation but also strongly demonstrates that our designed dynamic evolution circuit can accurately reproduce the time evolution behavior of the Ising model. This series of successful verifications under ideal conditions lays a solid foundation for our subsequent noisy simulations and the proposal and testing of error suppression methods.

[0131] (2) Performance evaluation of ZNE and NN-ZNE optimization

[0132] Based on the constructed composite noise model, we introduced noise into both single-bit and double-bit gates in the circuit. Noise was introduced to simulate the decoherence effect prevalent on real quantum devices. In this noisy environment, we systematically compared zero-noise extrapolation techniques using different extrapolation strategies, and focused on evaluating the performance of the NN-ZNE optimization scheme proposed in this invention.

[0133] We conducted noisy simulations and error suppression under three parameters (λ = 0.5, 0.9, 1.8) representing different physical regions, and compared the results obtained by different ZNE methods with the noise-free theoretical analytical solution. Specific results are presented in… Figure 5 .

[0134] These three sets of comparison graphs clearly show a significant systematic deviation between the original noisy data (marked by X points) and the theoretical analytical solution. Traditional ZNE methods, including linear, quadratic, cubic polynomial, and exponential function fitting methods, while mitigating noise-induced errors to some extent, still have limited correction effects, and their results still deviate significantly from the ideal curve. In stark contrast, the NN-ZNE method proposed in this invention (red curve) shows a high degree of agreement between its corrected evolution curve and the analytical solution under all test conditions, demonstrating correction accuracy and stability far exceeding traditional methods. Figure 5 Figures (a), (b), and (c) are shown in the diagram.

[0135] To quantitatively evaluate the performance of various methods, we calculated their percentage improvement in error relative to the original noisy data, such as... Figure 6 As shown in the figure, the NN-ZNE method shows a particularly significant improvement in overall performance, exceeding 80%. Cubic polynomial extrapolation performs second best, with an improvement of approximately 60%. It is noteworthy that linear and rational function extrapolations performed poorly in this test, with negative improvement percentages. This indicates that inappropriate extrapolation function selection is not only unhelpful but may even amplify errors, highlighting the risks inherent in traditional methods in application. After quantitative comparison, to more intuitively demonstrate the superiority of the NN-ZNE method, we directly compared it with the classic cubic function extrapolation fitting ZNE method, which performs best for each λ value, in the time evolution circuit results, as shown in the figure. Figure 7 As shown, regardless of whether it is in the ferromagnetic phase (λ=0.5), the critical region (λ=0.9), or the paramagnetic phase (λ=1.8), the error correction results of the NN-ZNE method (solid line) are consistently better than the best-performing classical ZNE method (dashed line), and more closely fit the ideal dynamic evolution trajectory. Finally, we examined the trend of the improvement effect of the NN-ZNE method with the system complexity (characterized by the λ value). Based on the ratio of the mean absolute error value of the results obtained by the NN-ZNE method to the mean absolute error value of the noisy results, we obtained the improvement factor of the NN-ZNE method. The results are plotted on [the graph]. Figure 8 The results show that the advantage of the NN-ZNE method becomes increasingly significant with increasing λ. When λ=0.5, the improvement factor is approximately 9.8 times; near the critical point at λ=0.9, this factor increases to 13.0 times; and in the strongly coupled system at λ=1.8, the improvement factor reaches 18.3 times. This trend clearly demonstrates that the NN-ZNE method possesses strong adaptability, especially in regions with stronger system interactions and more complex dynamic behavior, where its nonlinear modeling capability can more effectively learn and correct the complex errors introduced by noise.

[0136] To more quantitatively demonstrate the superiority of the NN-ZNE method, we present a comparison table here, summarizing the degree of noise improvement achieved by the NN-ZNE method and various classic ZNE fitting functions.

[0137] Table 1 summarizes the average performance improvement of different methods.

[0138]

[0139] In summary, the NN-ZNE scheme proposed in this invention comprehensively outperforms various traditional extrapolation methods in the noise suppression task of the Ising model dynamics evolution. This scheme demonstrates superior accuracy, stability, and robustness, strongly proving the enormous application potential of machine learning methods in the field of quantum computing error mitigation.

[0140] The preferred embodiments of the present application have been described above with reference to the accompanying drawings, but this does not limit the scope of the claims of the present application. Any modifications, equivalent substitutions, and improvements made by those skilled in the art without departing from the scope and substance of the embodiments of the present application shall be within the scope of the claims of the present application.

Claims

1. A method for modeling and optimizing composite noise in quantum circuits of NISQ devices, characterized in that: Includes the following steps: S1. In the quantum computing platform and Cqlib programming framework, a quantum circuit consisting of a ground state preparation circuit and a dynamic evolution circuit is constructed using the one-dimensional transverse field Ising model. A scalable composite noise model that comprehensively considers the coupling effect of decoherence and gate operation noise sources is inserted into the quantum circuit. The scalable composite noise model is constructed by integrating relaxation noise and phase noise through the Kraus operator using the Lindeblad master equation, and the generators of its dynamic evolution have additive properties. S2. Using the time scaling extrapolation method, the effective execution time of the quantum gate is adjusted by the noise scaling factor to generate the expected value of the quantum circuit under different noise intensities, i.e., the expected value of multiple noise levels. S3. Using the expected value of the multiple noise levels as input, a zero-noise extrapolation neural network is used for function fitting to output the predicted value under the zero-noise limit, thereby completing noise suppression; wherein: the zero-noise extrapolation neural network is formed by training the model parameters of a multilayer perceptron using the expected value of the multiple noise levels as training samples, the analytical solution of the one-dimensional transverse field Ising model as the supervision signal, and minimizing the mean square error as the objective; the multilayer perceptron includes an input layer, six hidden layers, and an output layer, with the number of neurons in the hidden layers decreasing in a descending order.

2. The method for modeling and optimizing composite noise in quantum circuits of NISQ devices according to claim 1, characterized in that: The quantum circuit is used to simulate the one-dimensional transverse field Ising model, and its Hamiltonian is: ， Quantum circuits achieve ground state preparation and dynamic evolution through Jordan-Wigner transform, Fourier transform, and Bogoliubov transform, and the circuit depth does not change with evolution time.

3. The method for modeling and optimizing composite noise in quantum circuits of NISQ devices according to claim 2, characterized in that: The relaxation noise The corresponding Kraus operator set is: ， The phase noise The corresponding Kraus operator set is: ， in, , , This represents the original execution time of the quantum gate.

4. The method for modeling and optimizing composite noise in quantum circuits of NISQ devices according to claim 3, characterized in that: The additivity is manifested as follows: ， in, For relaxation noise dissipation operators, For phase noise dissipation operators, For the system Hamiltonian, This is the system density matrix.

5. The method for modeling and optimizing composite noise in quantum circuits of NISQ devices according to claim 4, characterized in that: The time-scaling extrapolation method maintains the quantum gate pulse area constant and scales up. Stretch execution time And compress the amplitude And satisfy: , .

6. The method for modeling and optimizing composite noise in quantum circuits of NISQ devices according to claim 5, characterized in that: The multilayer perceptron employs the ReLU activation function, introduces Dropout layer regularization, and minimizes the loss function using the Adam optimization algorithm. ， in, For network parameter set, This is the analytical solution for the Ising model. This represents the number of training samples.

7. A noise suppression system for quantum circuits in NISQ devices, characterized in that: include: The quantum circuit construction module, within the quantum computing platform and Cqlib programming framework, utilizes the one-dimensional transverse field Ising model to construct quantum circuits consisting of ground state preparation circuits and dynamic evolution circuits. A scalable composite noise model, comprehensively considering the coupling effects of decoherence and gate operation noise sources, is inserted into the quantum circuits. Specifically, the scalable composite noise model is constructed using the Lindeblad master equation, integrating relaxation noise and phase noise through Kraus operators, and its dynamic evolution generators possess additive properties. The multi-noise data generation module uses a time scaling extrapolation method to generate the expected values ​​of quantum circuit observation under different noise intensities, i.e., multi-noise level expected values, by adjusting the effective execution time of the quantum gate through a noise scaling factor. The noise suppression execution module takes the expected values ​​of the multiple noise levels as input, performs function fitting through a zero-noise extrapolation neural network, and outputs the predicted value under the zero-noise limit to complete noise suppression. The zero-noise extrapolation neural network is trained by a neural network training module. This module uses the expected values ​​of the multiple noise levels as training samples, the analytical solution of the one-dimensional transverse field Ising model as the supervision signal, and minimizes the mean square error as the objective. It trains the model parameters of a multilayer perceptron to form the zero-noise extrapolation neural network. The multilayer perceptron includes an input layer, six hidden layers, and an output layer, with the number of neurons in the hidden layers decreasing in a descending order.

8. A noise suppression system for quantum circuits in an NISQ device according to claim 7, characterized in that: The coupling effect is achieved in a multi-qubit system by extending the single-bit noisy Kraus operator to the global Hilbert space through tensor product.

9. A computer-readable storage medium, characterized in that, The device contains a computer program that, when executed by a processor, implements the steps of a method for modeling and optimizing composite noise in a quantum circuit of an NISQ device as described in any one of claims 1-6.

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