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arXiv:2605.29242

Hybrid Gaussian-exponential zero-noise extrapolation for periodic circuits

Tao Wang, Yun Shang

quant-phSubmitted May 28, 2026• Announced May 29, 2026• 0 citations• Revised Jun 3, 2026 (v2)
v1 · May 28, 2026v2 · Jun 3, 2026
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Abstract

Zero-noise extrapolation provides a practical means of suppressing gate errors in current noisy intermediate-scale quantum hardware. The accuracy of the zero-noise estimate depends sensitively on the fidelity of the assumed noise model to the actual error scaling. This work introduces a hybrid Gaussian-exponential extrapolation scheme tailored for quantum circuits with periodic structure, which are ubiquitous in quantum algorithms. Under Pauli diagonal errors, by constructing and analyzing an approximate Markov process for the transfer of Pauli operators, we prove a central limit theorem: the noise amplification factor weakly approaches a log-normal distribution, which motivates augmenting the standard exponential model with Gaussian variance corrections. The resulting model requires no prior noise characterization and applies directly to arbitrary periodic circuits. Performance is assessed on Trotterized Ising dynamics, random circuits, and Grover search algorithm using Qiskit noise simulators. For moderate to large circuit depths, the hybrid model yields measurable reductions in bias relative to previous extrapolation variants, indicating its utility for error mitigation on near-term quantum hardware.

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Zero-noise extrapolation provides a practical means of suppressing gate errors in current noisy intermediate-scale quantum hardware. The accuracy of the zero-noise estimate depends sensitively on the fidelity of the assumed noise model to the actual error scaling. This work introduces a hybrid Gaussian-exponential extrapolation scheme tailored for quantum circuits with periodic structure, which are ubiquitous in quantum algorithms. Under Pauli diagonal errors, by constructing and analyzing an approximate Markov process for the transfer of Pauli operators, we prove a central limit theorem: the noise amplification factor weakly approaches a log-normal distribution, which motivates augmenting the standard exponential model with Gaussian variance corrections. The resulting model requires no prior noise characterization and applies directly to arbitrary periodic circuits. Performance is assessed on Trotterized Ising dynamics, random circuits, and Grover search algorithm using Qiskit noise simulators. For moderate to large circuit depths, the hybrid model yields measurable reductions in bias relative to previous extrapolation variants, indicating its utility for error mitigation on near-term quantum hardware.
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.
We investigate a hybrid quantum-classical approach to quantum error mitigation. We propose Classically Augmented Zero-Noise Extrapolation, a hybrid error-mitigation method in which high-noise Richardson extrapolation nodes are replaced by classically simulated estimates. These classical nodes have negligible sampling variance but introduce deterministic simulation bias. We derive the resulting variance reduction under optimal shot allocation and show that, for linear node spacings and fixed index cutoff, the coefficient-level reduction can be exponential. We validate the prediction numerically using Pauli-propagation simulations and demonstrate a reduction in mean-squared error when the truncation bias is sufficiently small.
Increasing the utility of currently available Noisy Intermediate-Scale Quantum (NISQ) devices requires developing efficient methods to mitigate hardware errors. In this work we propose a novel Cyclic Layout Permutations-based Zero-Noise Extrapolation (CLP-ZNE) protocol for such a task. The method leverages the inherent non-uniformity of gate errors in NISQ hardware to extrapolate the expectation value, averaged over cyclic circuit layout permutations, to the level of zero noise. In contrast to the previous layout permutation based approaches, for an nnn-qubit circuit CLP-ZNE requires execution of only O(n)O(n)O(n) and at most O(n2)O(n^2)O(n2) different circuit layouts for circuits of one-dimensional and arbitrary connectivity, respectively. When benchmarked against noise channels modeling the IBM Torino quantum computer, the method reduces a typical error in expectation values of n=12n=12n=12 qubit circuits by an order of magnitude, outperforming standard unitary folding ZNE. By demonstrating the ability to mitigate noise of real hardware specifications, including both depolarizing and T1/T2T_1/T_2T1​/T2​ relaxation processes, these results give evidence for the applicability of CLP-ZNE to present-day NISQ processors.
A common approach to deal with gate errors in modern quantum-computing hardware is zero-noise extrapolation. By artificially amplifying errors and extrapolating the expectation values obtained with different error strengths towards the zero-error (zero-noise) limit, the technique aims at rectifying errors in noisy quantum computing systems. For accurate extrapolation, it is essential to know the exact factors of the noise amplification. In this article, we propose a simple method for estimating the strength of errors occurring in a quantum circuit and demonstrate improved extrapolation results. The method determines the error strength for a circuit by appending to it the inverted circuit and measuring the probability of the initial state. The estimation of error strengths is easy to implement for arbitrary circuits and does not require previous characterization of noise properties. We compare this method with the conventional zero-noise extrapolation method and show that our method leads to a more accurate calculation of expectation values on current quantum hardware, showcasing its suitability for near-term quantum computing applications. Published by the American Physical Society 2024
Zero-noise extrapolation (ZNE) mitigates errors in near-term quantum devices by extrapolating measurements obtained at amplified noise levels to estimate noise-free expectation values. In practice, commonly used extrapolation models are fitted without enforcing physical constraints, which can yield predictions outside the valid range of quantum observables. In this work, we introduce physically bounded variants of polynomial, exponential, and polynomial--exponential extrapolation models by explicitly parameterizing the zero-noise estimate and constraining it during optimization. We evaluate the approach using a large synthetic benchmark comprising 180,000 circuits and approximately 3.6 million ZNE experiments generated under realistic device noise models derived from IBM quantum backends. We also perform preliminary validation on real quantum hardware using GHZ and W-state circuits. Across the synthetic benchmark, bounded extrapolation substantially reduces unphysical predictions and improves the stability of exponential- and polynomial--exponential-family models, whereas polynomial models show little difference between bounded and unbounded variants. Hardware experiments show similar qualitative behaviour: bounded models generally avoid pathological extrapolations and often provide a more reliable balance between accuracy and usable coverage. At the same time, the results highlight practical limitations of current devices, including stronger-than-expected noise effects and variability not fully captured by simulation models. These results suggest that enforcing physical constraints during extrapolation improves the reliability of ZNE and that this approach can be incorporated into existing workflows with minimal modification.
Abstract Quantum error mitigation is a key enabling technique for extracting reliable computational results from noisy intermediate-scale quantum devices. Among existing approaches, zero-noise extrapolation (ZNE) is widely employed, but its effectiveness deteriorates in high-noise regimes or deep circuits, where measurement outcomes converge to the random-guessing limit. To address this issue, we propose a lightweight, extrapolation-based error mitigation framework that leverages the intrinsic repetitive structure of quantum algorithms. We classify these algorithms into homogeneous and heterogeneous types and efficiently estimate block fidelities using shallow identity sequences of the core operational block, thereby avoiding global circuit folding. Experiments on IBM’s advanced superconducting platform and its corresponding simulator demonstrate that the proposed framework remains effective even in high-noise regimes where folding-based ZNE(f-ZNE) fails due to noise saturation. For a four-qubit Grover search, our approach restores the success probability to near its theoretical value, whereas f-ZNE yields negligible improvement. We further validate the proposed method on a 17-qubit quantum approximate optimization algorithm with layer-dependent parameters, demonstrating robustness beyond fixed-structure circuits. These results indicate that leveraging algorithmic structure provides a practical pathway to extending the performance of extrapolation-based error mitigation on current quantum processors.