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arXiv:2403.07025

Enhancing Quantum Variational Algorithms with Zero Noise Extrapolation via Neural Networks

Subhasree Bhattacharjee, Soumyadip Sarkar, Kunal Das, Bikramjit Sarkar

Published Mar 10, 2024• 0 citations
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Abstract

In the emergent realm of quantum computing, the Variational Quantum Eigensolver (VQE) stands out as a promising algorithm for solving complex quantum problems, especially in the noisy intermediate-scale quantum (NISQ) era. However, the ubiquitous presence of noise in quantum devices often limits the accuracy and reliability of VQE outcomes. This research introduces a novel approach to ameliorate this challenge by utilizing neural networks for zero noise extrapolation (ZNE) in VQE computations. By employing the Qiskit framework, we crafted parameterized quantum circuits using the RY-RZ ansatz and examined their behavior under varying levels of depolarizing noise. Our investigations spanned from determining the expectation values of a Hamiltonian, defined as a tensor product of Z operators, under different noise intensities to extracting the ground state energy. To bridge the observed outcomes under noise with the ideal noise-free scenario, we trained a Feed Forward Neural Network on the error probabilities and their associated expectation values. Remarkably, our model proficiently predicted the VQE outcome under hypothetical noise-free conditions. By juxtaposing the simulation results with real quantum device executions, we unveiled the discrepancies induced by noise and showcased the efficacy of our neural network-based ZNE technique in rectifying them. This integrative approach not only paves the way for enhanced accuracy in VQE computations on NISQ devices but also underlines the immense potential of hybrid quantum-classical paradigms in circumventing the challenges posed by quantum noise. Through this research, we envision a future where quantum algorithms can be reliably executed on noisy devices, bringing us one step closer to realizing the full potential of quantum computing.

DOI
10.48550/arxiv.2403.07025

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In the emergent realm of quantum computing, the Variational Quantum Eigensolver (VQE) stands out as a promising algorithm for solving complex quantum problems, especially in the noisy intermediate-scale quantum (NISQ) era. However, the ubiquitous presence of noise in quantum devices often limits the accuracy and reliability of VQE outcomes. This research introduces a novel approach to ameliorate this challenge by utilizing neural networks for zero noise extrapolation (ZNE) in VQE computations. By employing the Qiskit framework, we crafted parameterized quantum circuits using the RY-RZ ansatz and examined their behavior under varying levels of depolarizing noise. Our investigations spanned from determining the expectation values of a Hamiltonian, defined as a tensor product of Z operators, under different noise intensities to extracting the ground state energy. To bridge the observed outcomes under noise with the ideal noise-free scenario, we trained a Feed Forward Neural Network on the error probabilities and their associated expectation values. Remarkably, our model proficiently predicted the VQE outcome under hypothetical noise-free conditions. By juxtaposing the simulation results with real quantum device executions, we unveiled the discrepancies induced by noise and showcased the efficacy of our neural network-based ZNE technique in rectifying them. This integrative approach not only paves the way for enhanced accuracy in VQE computations on NISQ devices but also underlines the immense potential of hybrid quantum-classical paradigms in circumventing the challenges posed by quantum noise. Through this research, we envision a future where quantum algorithms can be reliably executed on noisy devices, bringing us one step closer to realizing the full potential of quantum computing.
In the noisy intermediate-scale quantum (NISQ) era, the Variational Quantum Eigensolver (VQE) comes out as an effective algorithm for addressing complex quantum challenges. However, the presence of noise in quantum devices frequently reduces the accuracy and reliability of VQE outcomes. This paper presents an innovative method to address this issue by using a neural network-based extrapolation technique in VQE calculation. By utilizing the Qiskit framework, we designed parameterized quantum circuits using the RY-RZ ansatz, and their performance was analyzed under different levels of depolarizing noise with bit-flip errors, phase-flip errors, and amplitude damping errors. Our inquiry involved analyzing the expected outcomes of a Hamiltonian across various levels of noise intensity with the objective of deducing the ground state energy (GSE). To connect the observed noisy outcomes with the ideal noise-free condition, a Feedforward Neural Network (FFNN) was trained using the error probabilities and their corresponding expectation values. This model accurately predicted the VQE results in an ideal noise-free scenario. Comparison of the result of simulation and real quantum hardware executions revealed noise-induced inconsistencies, highlighting the effectiveness of this neural network-based extrapolation approach in correcting them. This comprehensive method improves the accuracy of VQE calculation on NISQ devices and highlights the significant potential of blending quantum and classical methods to address the threats imposed by quantum noise. The comparison of results between FFNN, convolutional neural network (CNN), and long short-term memory (LSTM) network reveals that FFNN predicts outcomes with more accuracy but in less time.
We introduce a general framework called neural-network- (NN) encoded variational quantum algorithms (VQAs), or NNVQA for short, to address the challenges of implementing VQAs on noisy intermediate-scale quantum (NISQ) computers. Specifically, NNVQA feeds input (such as parameters of a Hamiltonian) from a given problem to a neural network and uses its outputs to parameterize an ansatz circuit for the standard VQA. Combining the strengths of NN and parameterized quantum circuits, NNVQA can accelerate the training process of VQAs and handle a broad family of related problems with varying input parameters with the pretrained NN. To concretely illustrate the merits of NNVQA, we present results on a NN variational quantum eigensolver (VQE) for solving the ground state of parameterized XXZXXZXXZ spin models in one and two dimensions. Our results demonstrate that NNVQE is able to estimate the ground-state energies of parameterized Hamiltonians with high precision without fine tuning, and significantly reduce the overall training cost to estimate ground-state properties across the phases of the given Hamiltonian. We also employ an active learning strategy to further increase the training efficiency while maintaining prediction accuracy. These encouraging results demonstrate that NNVQAs offer an alternative hybrid quantum-classical paradigm to utilize NISQ resources for solving more realistic and challenging computational problems.
As a hybrid quantum-classical algorithm, the variational quantum eigensolver is widely applied in quantum chemistry simulations, especially in computing the electronic structure of complex molecular systems. However, on existing noisy intermediate-scale quantum devices, some factors such as quantum decoherence, measurement errors, and gate operation imprecisions are unavoidable. To overcome these challenges, this study proposes an efficient noise-mitigating variational quantum eigensolver for accurate computation of molecular ground state energies in noisy environments. We design the quantum circuit with reference to the structure of matrix product states and utilize it to pre-train the circuit parameters, which ensures circuit stability and mitigates fluctuations caused by initialization. We also employ zero-noise extrapolation to mitigate quantum noise and combine it with neural networks to improve the accuracy of the noise-fitting function, which significantly eliminates noise interference. Furthermore, we implement an intelligent grouping strategy for measuring Hamiltonian Pauli strings, which not only reduces measurement errors but also improves sampling efficiency. We perform numerical simulations to solve the ground state energy of the H4molecule by using MindSpore Quantum framework, and the results demonstrate that our algorithm can constrain noise errors within the range of O(10−2)∼O(10−1)\mathcal{O}\left( {{{10}^{ - 2}}} \right)\sim \mathcal{O}\left( {{{10}^{ - 1}}} \right)O(10−2)∼O(10−1), outperforming mainstream variational quantum eigensolvers. This work provides a new strategy for high-precision quantum chemistry calculations on near-term noisy quantum hardware.
The execution of Variational Quantum Algorithms on Noisy Intermediate-Scale Quantum hardware is severely impacted by hardware noise variations over time and decoherence. This work introduces an Adaptive Noise-Aware Variational Quantum Classifier (ANAVQC) that integrates Zero-Noise Extrapolation (ZNE) dynamically into the training loop to counter variations. The framework approximates the instantaneous depolarizing noise from calibration and adapts noise scaling and linear extrapolation to estimate noise-free expectation values at training. A two-qubit parameterized quantum circuit is built and simulated in MATLAB, with the inclusion of stochastic gradient descent coupled with parameter-shift-based gradients. The model is conditioned on a synthetic binary classification data under a time-varying depolarizing noise profile and exhibits improved convergence stability and classification accuracy over non-adaptive methods. The simulation results identify that adaptive ZNE increases model robustness against time-varying noise and offers a scalable route to reliable quantum machine learning on practical NISQ hardware. The current ANAVQC scheme is extendable to multi-qubit platforms and practical quantum backends for the purposes of future experimental testing.
The variational quantum eigensolver (VQE) is a widely employed method to solve electronic structure problems in the current noisy intermediate-scale quantum (NISQ) devices. However, due to inherent noise in the NISQ devices, VQE results on NISQ devices often deviate significantly from the results obtained on noiseless statevector simulators or traditional classical computers. The iterative nature of VQE further amplifies the errors in each loop. Recent works have explored ways to integrate deep neural networks (DNN) with VQE to mitigate iterative errors, albeit primarily limited to the noiseless statevector simulators. In this work, we trained DNN models across various quantum circuits and examined the potential of two DNN-VQE approaches, DNN1 and DNNF, for predicting the ground state energies of small molecules in the presence of device noise. We carefully examined the accuracy of the DNN1, DNNF, and VQE methods on both noisy simulators and real quantum devices by considering different ansatzes of varying qubit counts and circuit depths. Our results illustrate the advantages and limitations of both VQE and DNN-VQE approaches. Notably, both DNN1 and DNNF methods consistently outperform the standard VQE method in providing more accurate ground state energies in noisy environments. However, despite being more accurate than VQE, the energies predicted using these methods on real quantum hardware remain meaningful only at reasonable circuit depths (depth = 15, gates = 21). At higher depths (depth = 83, gates = 112), they deviate significantly from the exact results. Additionally, we find that DNNF does not offer any notable advantage over VQE in terms of speed. Consequently, our study recommends DNN1 as the preferred method for obtaining quick and accurate ground state energies of molecules on current quantum hardware, particularly for quantum circuits with lower depth and fewer qubits.
Performance Optimization of Noise-Mitigated Quantum-Classical Hybrid Classifiers via Zero-Noise Extrapolation in NISQ Environments Overview Near-Term Intermediate-Scale Quantum (NISQ) hardware remains heavily constrained by physical noise, gate infidelities, and decoherence, which severely degrade the performance of Variational Quantum Classifiers (VQCs). This research presents an end-to-end, noise-mitigated hybrid quantum-classical machine learning pipeline that integrates PyTorch classical layers with PennyLane parameterized quantum circuits (PQCs) and applies Zero-Noise Extrapolation (ZNE) via Mitiq to counteract hardware noise. Key Highlights & Contributions Modular Hybrid Architecture: Combines classical feature mapping and post-processing in PyTorch with parameterized quantum variational circuits built in PennyLane. Digital Error Mitigation: Evaluates global gate folding (G→GG†GG \to G G^\dagger GG→GG†G) across varying scale factors (λ∈{1,3,5}\lambda \in \{1, 3, 5\}λ∈{1,3,5}) to extrapolate expectation values back to the zero-noise limit. Accuracy Recovery: Demonstrates that ZNE recovers classification accuracy near ideal noiseless performance (98.5%98.5\%98.5%) under low-to-moderate depolarizing noise levels (p < 0.01), outperforming unmitigated noisy executions by up to 11.3%. Model Benchmarking: Directly compares linear and second-order polynomial extrapolation models, analyzing variance trade-offs under finite shot noise constraints. Repository & Code Contents This deposit includes the full PDF manuscript alongside the accompanying code archive: Noise_Mitigated_Quantum_Classical_Hybrid_Classifier_with_Zero_Noise_Extrapolation.pdf — Full IEEE-formatted research paper. Noise-MitigatedQuantum-ClassicalHybridClassifierwithZero-NoiseExtrapolation.ipynb — Executable Jupyter Notebook containing the full PyTorch–PennyLane–Mitiq benchmarking pipeline. requirements.txt & README — Environment setup and reproduction instructions. How to Cite If you find this work or codebase helpful for your research in Quantum Machine Learning (QML) or Quantum Error Mitigation (QEM), please cite this preprint using the Zenodo DOI provided above.