Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.15014

Hybrid physical/logical zero-noise extrapolation with limited logical executions

D. V. Babukhin, W. V. Pogosov

quant-phSubmitted Apr 16, 2026• Announced Apr 17, 2026• 0 citations• Revised Jun 4, 2026 (v2)
v1 · Apr 16, 2026v2 · Jun 4, 2026v3 · Jul 20, 2026
arXivHTMLPDF

Abstract

Before full fault tolerance, partially error-corrected logical executions may be available but costly. We formulate zero-noise extrapolation as a statistical-design problem in which the physical/logical execution mode is an extrapolation variable. Logical circuits provide low-noise anchors, while physical folded circuits supply a cheaper noise lever arm. In a calibrated $p_L=γp$ model, we derive Richardson variance prefactors, folded-circuit runtime ratios, optimal shot allocation, and bias--variance criteria. We demonstrate that for $γ\lesssim0.1$, the mixed strategy can substantially reduce runtime under the stated resource model.

Comments
31 pages, 7 figures

Related papers

Zero Noise Extrapolation on Logical Qubits by Scaling the Error Correction Code Distance

Misty Wahl, Andrea Mari, Nathan Shammah +2 more

In this work, we migrate the quantum error mitigation technique of Zero-Noise Extrapolation (ZNE) to fault-tolerant quantum computing. We employ ZNE on logically encoded qubits rather than physical qubits. This approach will be useful in a regime where quantum error correction (QEC) is implementable but the number of qubits available for QEC is limited. Apart from illustrating the utility of a traditional ZNE approach (circuit-level unitary folding) for the QEC regime, we propose a novel noise scaling ZNE method specifically tailored to QEC: distance scaled ZNE (DS-ZNE). DS-ZNE scales the distance of the error correction code, and thereby the resulting logical error rate, and utilizes this code distance as the scaling ‘knob’ for ZNE. Logical qubit error rates are scaled until the maximum achievable code distance for a fixed number of physical qubits, and lower error rates (i.e., effectively higher code distances) are achieved via extrapolation techniques migrated from traditional ZNE. Furthermore, to maximize physical qubit utilization over the ZNE experiments, logical executions at code distances lower than the maximum allowed by the physical qubits on the quantum device are executed in parallel across the device, thereby reducing overall circuit execution costs. We validate our proposal with numerical simulation for the surface code and confirm that ZNE lowers the logical error rates and increases the effective code distance beyond the physical capability of the quantum device. For instance, at a physical code distance of 11, the DS-ZNE effective code distance is 17, and at a physical code distance of 13, the DS-ZNE effective code distance is 21. When the proposed technique is compared against unitary folding ZNE under the constraint of a fixed number of executions of the quantum device, DS-ZNE outperforms unitary folding by up to 92% in terms of the post-ZNE logical error rate.

arXiv:2304.14985•IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 17, 2023•13 citations

Classically Augmented Zero-Noise Extrapolation

Timon Scheiber

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.

quant-pharXiv:2607.25746•Jul 29, 2026•0 citations

Folding-Free Zero-Noise Extrapolation by Layout-induced Noise Diversity

Debarthi Pal, Yogesh Simmhan

Near term quantum processors operate in a noise dominated regime, motivating error mitigation techniques that recover accurate expectation values without full fault tolerance. Zero Noise Extrapolation (ZNE) is a widely used but biased error mitigation method that lacks rigorous error bounds. Its effective application requires nontrivial technical choices, most notably the selection of noise scaling factors and extrapolation models, making ZNE sensitive to user expertise and often necessitating costly trial and error procedures. Here, we introduce Folding Free Zero Noise Extrapolation (FF-ZNE), a method that removes the need for noise factor selection by achieving effective noise amplification without circuit folding. FF-ZNE exploits isomorphic hardware layouts with distinct native noise profiles, such that executing a fixed circuit across these layouts induces controllable variations in the effective noise strength. Under a depolarizing noise model, we analytically show that the resulting extrapolation admits a fixed linear form, eliminating extrapolator choice and enabling a seamless, user independent mitigation procedure. We further propose two algorithms that identify sets of isomorphic hardware layouts on which a given circuit yields sufficiently distinct expectation values to enable reliable zero noise extrapolation. Experiments on a 133 qubit IBM Quantum device demonstrate that FF-ZNE yields mitigated expectation values with average deviations of ~6% and 4.5% for up to 50 qubit EfficientSU2 (sparse) and Hamiltonian simulation (dense) circuits, respectively. The method is thus scalable and applicable to a broad range of circuits. By eliminating noise factor and extrapolator selection, FF-ZNE transforms zero noise extrapolation from a technique requiring expert tuning into a practical, scalable, and broadly accessible error mitigation method for current quantum hardware.

quant-pharXiv:2603.13949•Mar 17, 2026•0 citations

Hybrid Gaussian-exponential zero-noise extrapolation for periodic circuits

Tao Wang, Yun Shang

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.

quant-pharXiv:2605.29242•May 29, 2026•0 citations

Improving Zero-Noise Extrapolation via Physically Bounded Models

Andriy Miranskyy, Adam Sorrenti, Jasmine Thind +1 more

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.

quant-pharXiv:2604.24475•Apr 28, 2026•0 citations

Quantum Noise Mitigation with Adaptive Zero-Noise Extrapolation: A Contextual Multi-Armed Bandits Approach

Ratun Rahman, Dinh C. Nguyen

Variational quantum circuits (VQCs) are central to near-term quantum computing, yet their practical deployment is severely hindered by noise. While existing error mitigation methods, such as zero-noise extrapolation (ZNE), typically assume static noise, real noisy intermediate-scale quantum (NISQ) systems exhibit dynamic, time-varying noise that remains largely unaddressed. To overcome this critical gap, our work introduces a novel adaptive noise mitigation framework for VQCs that integrates ZNE with contextual multi-armed bandits (CMAB), enabling dynamic, context-aware selection of circuit-folding levels based on ansatz parameters (e.g., depth, parameter count) and the evolving noise environment. Unlike fixed-fold or heuristic ZNE, our approach uses online adaptation to improve the accuracy of ZNE and reduce redundant quantum circuit executions. Our extensive simulations and experiments on real quantum hardware reveal the following important properties: (i) deeper VQCs accumulate noise, degrading accuracy and increasing the number of quantum circuit executions; (ii) ZNE restores estimator fidelity when the folding level is chosen appropriately; and (iii) CMAB-guided folding cuts quantum circuit execution round trips by up to 40\%, bytes exchanged by up to 35\%, and end-to-end cost by up to 30\% under a 10~Mbps budget, with up to 6.9\% higher estimator fidelity (CIFAR-10, depth 3, noise band $η=0.05$), versus fixed-fold and grid-search ZNE. These results demonstrate substantial performance gains over existing noise mitigation methods, underscoring the effectiveness of our design in supporting robust noise mitigation for VQCs. The source code is also publicly released to support reproducibility.

quant-pharXiv:2608.06426•Aug 7, 2026•0 citations
Before full fault tolerance, partially error-corrected logical executions may be available but costly. We formulate zero-noise extrapolation as a statistical-design problem in which the physical/logical execution mode is an extrapolation variable. Logical circuits provide low-noise anchors, while physical folded circuits supply a cheaper noise lever arm. In a calibrated pL=γpp_L=γppL​=γp model, we derive Richardson variance prefactors, folded-circuit runtime ratios, optimal shot allocation, and bias--variance criteria. We demonstrate that for γ≲0.1γ\lesssim0.1γ≲0.1, the mixed strategy can substantially reduce runtime under the stated resource model.
In this work, we migrate the quantum error mitigation technique of Zero-Noise Extrapolation (ZNE) to fault-tolerant quantum computing. We employ ZNE on logically encoded qubits rather than physical qubits. This approach will be useful in a regime where quantum error correction (QEC) is implementable but the number of qubits available for QEC is limited. Apart from illustrating the utility of a traditional ZNE approach (circuit-level unitary folding) for the QEC regime, we propose a novel noise scaling ZNE method specifically tailored to QEC: distance scaled ZNE (DS-ZNE). DS-ZNE scales the distance of the error correction code, and thereby the resulting logical error rate, and utilizes this code distance as the scaling ‘knob’ for ZNE. Logical qubit error rates are scaled until the maximum achievable code distance for a fixed number of physical qubits, and lower error rates (i.e., effectively higher code distances) are achieved via extrapolation techniques migrated from traditional ZNE. Furthermore, to maximize physical qubit utilization over the ZNE experiments, logical executions at code distances lower than the maximum allowed by the physical qubits on the quantum device are executed in parallel across the device, thereby reducing overall circuit execution costs. We validate our proposal with numerical simulation for the surface code and confirm that ZNE lowers the logical error rates and increases the effective code distance beyond the physical capability of the quantum device. For instance, at a physical code distance of 11, the DS-ZNE effective code distance is 17, and at a physical code distance of 13, the DS-ZNE effective code distance is 21. When the proposed technique is compared against unitary folding ZNE under the constraint of a fixed number of executions of the quantum device, DS-ZNE outperforms unitary folding by up to 92% in terms of the post-ZNE logical error rate.
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.
Near term quantum processors operate in a noise dominated regime, motivating error mitigation techniques that recover accurate expectation values without full fault tolerance. Zero Noise Extrapolation (ZNE) is a widely used but biased error mitigation method that lacks rigorous error bounds. Its effective application requires nontrivial technical choices, most notably the selection of noise scaling factors and extrapolation models, making ZNE sensitive to user expertise and often necessitating costly trial and error procedures. Here, we introduce Folding Free Zero Noise Extrapolation (FF-ZNE), a method that removes the need for noise factor selection by achieving effective noise amplification without circuit folding. FF-ZNE exploits isomorphic hardware layouts with distinct native noise profiles, such that executing a fixed circuit across these layouts induces controllable variations in the effective noise strength. Under a depolarizing noise model, we analytically show that the resulting extrapolation admits a fixed linear form, eliminating extrapolator choice and enabling a seamless, user independent mitigation procedure. We further propose two algorithms that identify sets of isomorphic hardware layouts on which a given circuit yields sufficiently distinct expectation values to enable reliable zero noise extrapolation. Experiments on a 133 qubit IBM Quantum device demonstrate that FF-ZNE yields mitigated expectation values with average deviations of ~6% and 4.5% for up to 50 qubit EfficientSU2 (sparse) and Hamiltonian simulation (dense) circuits, respectively. The method is thus scalable and applicable to a broad range of circuits. By eliminating noise factor and extrapolator selection, FF-ZNE transforms zero noise extrapolation from a technique requiring expert tuning into a practical, scalable, and broadly accessible error mitigation method for current quantum hardware.
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.
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.
Variational quantum circuits (VQCs) are central to near-term quantum computing, yet their practical deployment is severely hindered by noise. While existing error mitigation methods, such as zero-noise extrapolation (ZNE), typically assume static noise, real noisy intermediate-scale quantum (NISQ) systems exhibit dynamic, time-varying noise that remains largely unaddressed. To overcome this critical gap, our work introduces a novel adaptive noise mitigation framework for VQCs that integrates ZNE with contextual multi-armed bandits (CMAB), enabling dynamic, context-aware selection of circuit-folding levels based on ansatz parameters (e.g., depth, parameter count) and the evolving noise environment. Unlike fixed-fold or heuristic ZNE, our approach uses online adaptation to improve the accuracy of ZNE and reduce redundant quantum circuit executions. Our extensive simulations and experiments on real quantum hardware reveal the following important properties: (i) deeper VQCs accumulate noise, degrading accuracy and increasing the number of quantum circuit executions; (ii) ZNE restores estimator fidelity when the folding level is chosen appropriately; and (iii) CMAB-guided folding cuts quantum circuit execution round trips by up to 40\%, bytes exchanged by up to 35\%, and end-to-end cost by up to 30\% under a 10~Mbps budget, with up to 6.9\% higher estimator fidelity (CIFAR-10, depth 3, noise band η=0.05η=0.05η=0.05), versus fixed-fold and grid-search ZNE. These results demonstrate substantial performance gains over existing noise mitigation methods, underscoring the effectiveness of our design in supporting robust noise mitigation for VQCs. The source code is also publicly released to support reproducibility.