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IEEE Access

Input-Data Information Preservation and Protocol-Specific Error Mitigation in Noisy Fixed-Parameter Quantum Front Ends: A Reproducible Simulation Study

Hongmin Jung

Published Jan 1, 2026• 0 citations
Publisher

Abstract

I examine whether a noisy fixed-parameter quantum front end preserves class information from input data before variational optimization. Only the input vector $x$ is encoded; the label is used after circuit execution to estimate training distributions and score disjoint held-out inputs. The main density-matrix grid contains 10,800 directly simulated configurations over two binary tasks, widths $n\in \{2,3,4\}$ , every integer depth $L=1,\ldots,12$ , five depolarizing strengths, and 30 circuit seeds. At $n=4$ , mean held-out mutual information has a finite-depth maximum that shifts from $L=6$ to 5 for Gaussian data and from $L=7$ to 6 for the Wisconsin Diagnostic Breast Cancer (WDBC) benchmark as $p$ increases from 0 to 0.04. In five-fold complete-seed validation, including training-distribution mutual information with declared design variables raises held-out-accuracy $R^{2}$ from 0.724 to 0.961 for Gaussian data and from 0.714 to 0.860 for WDBC. Training total variation is the stronger binary comparator in complete-seed validation; test-set total variation is reported only as a non-deployable post-outcome oracle. Paired seed-level RMSE improvements from mutual information are 0.0185 (95% bootstrap interval 0.0179–0.0190) and 0.00634 (0.00592–0.00675), and finite-shot training estimates retain most of that gain. Twenty independently seeded Gaussian realizations, twenty repeated WDBC partitions, per-input 256-shot tests, and explicit rotation-schedule ablations expose both transfer and limitations. A family-matched damped-alignment response outperforms an equal-parameter-count polynomial comparator under identical grouped folds but transfers weakly to some widths, encodings, and schedules. Explicit local-folding zero-noise extrapolation and tensor-product measurement-error mitigation improve information and held-out likelihood accuracy in selected simulated budgets; small-cluster sensitivity and failure cases are retained. The accompanying reproducibility archive contains the complete source, raw counts, parameter registries, 2,000-replicate bootstrap vectors, mitigation replicates, automated tests, and resumable one-click workflow. I make no hardware, trained-VQNN, clinical, quantum-advantage, or universal-scaling claim.

DOI
10.1109/access.2026.3732383

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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

Certification cost of quantum models: measurement correlation, not parameter count

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Reporting the Fisher geometry of a trained variational quantum model is routine; quoting the shot budget that would establish it is not. Certifying an empirical Fisher matrix to relative Frobenius error $\varepsilon$ under coordinate-wise parameter shift costs $Θ(B p^{2} V/(\varepsilon^{2} G))$ circuit executions, where $V$ is the measured readout variance and $G$ the measured squared gradient norm, with uniform allocation optimal in that class. One constant reproduces the cost of two circuit families whose exponents differ by a full power of $p$. The exponent is an identity in how $nV$ and $nG$ scale with the register, holding family by family to $0.001$ across 624 matrix-product-state cells once the finite-$p$ prefactor is removed. The cubic cost is therefore a finite-size window, set by whether the readout light cone grows with the register. A product family to 256 qubits gives $1.966$ (95% CI $1.934$--$1.997$); a brickwork entangler falls from $2.853$ below ten qubits to $1.715$ beyond sixty-four; a blocked entangler gives $1.984$ at a fixed cone width against $3.034$ at a proportional one. Fixed device connectivity fixes the cone, so a cubic budget from a small simulation overestimates a large machine, on top of hardware multipliers $2.07\times$ ($1.41$--$3.02$), $2.38\times$ and $1.91\times$ on ibm_marrakesh, ibm_fez and ibm_kingston. Cost-optimal readout weights cut the measured shot budget by $2.67\times$ ($1.33$--$4.00$) on hardware, flat from four to twelve qubits. A discrepancy model fitted on cheap circuits transfers its mean inside the calibration grid and, at six larger sizes named before the data, does not: nominal 90% intervals cover 36%, and split conformal is the only rung that stays near nominal.

quant-pharXiv:2609.14424•Sep 15, 2026•0 citations

When Does Zero-Noise Extrapolation Help? A Controlled Bias–Variance Analysis of Error Mitigation for Variational Quantum Chemistry

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Variational Quantum Eigensolvers (VQE) promise molecular ground-state energies on today's noisy intermediate-scale quantum (NISQ) processors, but device noise biases the estimated energy severely — under a device-calibrated noise model, the hydrogen molecule's energy error grows from 0.000 mHa (noiseless) to 303.6 mHa (noisy), roughly 190× the "chemical accuracy" threshold of 1.6 mHa that computational chemistry requires. Error-mitigation techniques repair such biases in software, but published results usually report each technique in isolation, making it impossible to know which mitigation to buy with a fixed shot budget. This work answers that question with a controlled comparison. Under identical circuits, noise profiles and shot budgets, I measure the bias and variance of four estimator stacks — raw sampling, twirled-readout extinction (TREX-style calibrated readout inversion), zero-noise extrapolation (ZNE), and ZNE+TREX — across independent random seeds. Three findings emerge. (1) ZNE is a bias–variance tradeoff, not a free lunch: a 7-point Richardson extrapolant reduces the deterministic bias 3200-fold (303.6 → 0.094 mHa) but amplifies statistical noise by a measured factor of up to 25× per run, while TREX removes readout bias with no variance penalty. (2) ZNE has a finite operating window in circuit depth: error recovery falls 99.97% → 99.1% → 95.5% as CX count grows 56 → 112 → 168, and is exhausted entirely for a 1616-CX lithium-hydride circuit at near-term error rates. (3) A noise-bias-tailored repetition code corrects the dominant error axis with logical error rates matching theory (3p², 10p³, 35p⁴ for distances 3, 5, 7) at strictly linear gate overhead (gates = 6d − 6). The results reframe mitigation selection as budget allocation on an empirically measured bias–variance frontier and provide open-source tooling to reproduce every number.

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Machine Learning-based Quantum Error Mitigation for Variational Algorithms

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Machine Learning-based quantum error mitigation (ML-QEM) has emerged as a promising approach for improving the performance of noisy quantum algorithms. However, existing ML-QEM methods often have restricted applicability to variational circuits and rely on inaccessible noiseless training data. In this work, we propose a practical ML-QEM protocol tailored to variational quantum algorithms, which generates training data by simulating (near-)Clifford circuits. This data is used for model selection and training, producing a mitigation model that can correct variational circuits with arbitrary parameters and transfer across different target Hamiltonians of similar structure. We benchmark the proposed method on the Variational Quantum Eigensolver (VQE) task for the Sherrington-Kirkpatrick Hamiltonian of up to $n=12$ qubits under various noise models, analyzing its effect on trainability and comparing its performance against standard Zero-Noise Extrapolation (ZNE). The results demonstrate consistent several-fold error suppression across all tested settings and superior performance over ZNE in the high-noise regime, providing evidence for the applicability of the proposed protocol to present-day NISQ processors.

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Noise-Mitigated Quantum-Classical Hybrid Classification via Zero-Noise Extrapolation

Alexandru-Florin Boboc

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 ($G \to G G^\dagger G$) across varying scale factors ($\lambda \in \{1, 3, 5\}$) to extrapolate expectation values back to the zero-noise limit. Accuracy Recovery: Demonstrates that ZNE recovers classification accuracy near ideal noiseless performance ($98.5\%$) 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.

•Aug 10, 2026•0 citations

Budget-Aware Neural Error Mitigation forVariational Quantum Algorithms

Guilin Zhang, Kai Zhao, Xu Chu

Abstract Variational quantum algorithms (VQAs) are limited by device noise, and error mitigation is the standard remedy on near-term hardware. Choosing a mitigation method for deployment requires comparing its true cost, yet the usual comparison hides it: it fixes the number of shots per circuit execution, while the methods differ sharply in how many executions they need per estimate. Zero-noise extrapolation (ZNE) needs three to five, Clifford data regression (CDR) more than thirty per target circuit, and a trained neural model exactly one; ZNE's accuracy further rests on a noise-scaling assumption that real devices violate. Inside a variational loop that evaluates thousands of parameter settings, it is this execution multiplicity, not the accuracy at unlimited shots, that decides which method is affordable. Our premise is that the comparison that matters is at equal total shot budget per evaluation point. We present budget-conditioned neural error mitigation (NEM): one network corrects a single shot-limited measurement from circuit and device features, conditioned on the shot-noise scale so that one model serves every budget. We evaluate every method—including each ZNE fold point and each CDR training circuit—under a protocol that spends the same total shots per evaluation point. Under it, NEM is the most accurate method at every budget under coherent miscalibration (reducing error by 72–81%), and the only method that beats unmitigated execution at budgets up to 2^14 shots under stochastic device noise (11–20%); under miscalibration CDR is also positive but overtakes NEM only at 2^16 shots, at 32x the per-evaluation cost. We further characterize when extrapolation fails: under control-offset miscalibration, folding ZNE returns a worse answer than no mitigation on 38–63% of instances in simulation, and on up to 88% on a real IBM Heron processor. On that processor, fine-tuning the model on a small on-device set—120 circuits with classically computed labels—narrows the simulation-to-hardware gap, recovering a 39% error reduction on held-out device instances from the same calibration epoch. Improvements hold from 4 to 20 qubits. Budget-aware accounting, we argue, is the comparison that matters for deployment.

•Aug 14, 2026•0 citations
I examine whether a noisy fixed-parameter quantum front end preserves class information from input data before variational optimization. Only the input vector xxx is encoded; the label is used after circuit execution to estimate training distributions and score disjoint held-out inputs. The main density-matrix grid contains 10,800 directly simulated configurations over two binary tasks, widths n∈{2,3,4}n\in \{2,3,4\}n∈{2,3,4} , every integer depth L=1,…,12L=1,\ldots,12L=1,…,12 , five depolarizing strengths, and 30 circuit seeds. At n=4n=4n=4 , mean held-out mutual information has a finite-depth maximum that shifts from L=6L=6L=6 to 5 for Gaussian data and from L=7L=7L=7 to 6 for the Wisconsin Diagnostic Breast Cancer (WDBC) benchmark as ppp increases from 0 to 0.04. In five-fold complete-seed validation, including training-distribution mutual information with declared design variables raises held-out-accuracy R2R^{2}R2 from 0.724 to 0.961 for Gaussian data and from 0.714 to 0.860 for WDBC. Training total variation is the stronger binary comparator in complete-seed validation; test-set total variation is reported only as a non-deployable post-outcome oracle. Paired seed-level RMSE improvements from mutual information are 0.0185 (95% bootstrap interval 0.0179–0.0190) and 0.00634 (0.00592–0.00675), and finite-shot training estimates retain most of that gain. Twenty independently seeded Gaussian realizations, twenty repeated WDBC partitions, per-input 256-shot tests, and explicit rotation-schedule ablations expose both transfer and limitations. A family-matched damped-alignment response outperforms an equal-parameter-count polynomial comparator under identical grouped folds but transfers weakly to some widths, encodings, and schedules. Explicit local-folding zero-noise extrapolation and tensor-product measurement-error mitigation improve information and held-out likelihood accuracy in selected simulated budgets; small-cluster sensitivity and failure cases are retained. The accompanying reproducibility archive contains the complete source, raw counts, parameter registries, 2,000-replicate bootstrap vectors, mitigation replicates, automated tests, and resumable one-click workflow. I make no hardware, trained-VQNN, clinical, quantum-advantage, or universal-scaling claim.
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.
Reporting the Fisher geometry of a trained variational quantum model is routine; quoting the shot budget that would establish it is not. Certifying an empirical Fisher matrix to relative Frobenius error ε\varepsilonε under coordinate-wise parameter shift costs Θ(Bp2V/(ε2G))Θ(B p^{2} V/(\varepsilon^{2} G))Θ(Bp2V/(ε2G)) circuit executions, where VVV is the measured readout variance and GGG the measured squared gradient norm, with uniform allocation optimal in that class. One constant reproduces the cost of two circuit families whose exponents differ by a full power of ppp. The exponent is an identity in how nVnVnV and nGnGnG scale with the register, holding family by family to 0.0010.0010.001 across 624 matrix-product-state cells once the finite-ppp prefactor is removed. The cubic cost is therefore a finite-size window, set by whether the readout light cone grows with the register. A product family to 256 qubits gives 1.9661.9661.966 (95% CI 1.9341.9341.934--1.9971.9971.997); a brickwork entangler falls from 2.8532.8532.853 below ten qubits to 1.7151.7151.715 beyond sixty-four; a blocked entangler gives 1.9841.9841.984 at a fixed cone width against 3.0343.0343.034 at a proportional one. Fixed device connectivity fixes the cone, so a cubic budget from a small simulation overestimates a large machine, on top of hardware multipliers 2.07×2.07\times2.07× (1.411.411.41--3.023.023.02), 2.38×2.38\times2.38× and 1.91×1.91\times1.91× on ibm_marrakesh, ibm_fez and ibm_kingston. Cost-optimal readout weights cut the measured shot budget by 2.67×2.67\times2.67× (1.331.331.33--4.004.004.00) on hardware, flat from four to twelve qubits. A discrepancy model fitted on cheap circuits transfers its mean inside the calibration grid and, at six larger sizes named before the data, does not: nominal 90% intervals cover 36%, and split conformal is the only rung that stays near nominal.
Variational Quantum Eigensolvers (VQE) promise molecular ground-state energies on today's noisy intermediate-scale quantum (NISQ) processors, but device noise biases the estimated energy severely — under a device-calibrated noise model, the hydrogen molecule's energy error grows from 0.000 mHa (noiseless) to 303.6 mHa (noisy), roughly 190× the "chemical accuracy" threshold of 1.6 mHa that computational chemistry requires. Error-mitigation techniques repair such biases in software, but published results usually report each technique in isolation, making it impossible to know which mitigation to buy with a fixed shot budget. This work answers that question with a controlled comparison. Under identical circuits, noise profiles and shot budgets, I measure the bias and variance of four estimator stacks — raw sampling, twirled-readout extinction (TREX-style calibrated readout inversion), zero-noise extrapolation (ZNE), and ZNE+TREX — across independent random seeds. Three findings emerge. (1) ZNE is a bias–variance tradeoff, not a free lunch: a 7-point Richardson extrapolant reduces the deterministic bias 3200-fold (303.6 → 0.094 mHa) but amplifies statistical noise by a measured factor of up to 25× per run, while TREX removes readout bias with no variance penalty. (2) ZNE has a finite operating window in circuit depth: error recovery falls 99.97% → 99.1% → 95.5% as CX count grows 56 → 112 → 168, and is exhausted entirely for a 1616-CX lithium-hydride circuit at near-term error rates. (3) A noise-bias-tailored repetition code corrects the dominant error axis with logical error rates matching theory (3p², 10p³, 35p⁴ for distances 3, 5, 7) at strictly linear gate overhead (gates = 6d − 6). The results reframe mitigation selection as budget allocation on an empirically measured bias–variance frontier and provide open-source tooling to reproduce every number.
Machine Learning-based quantum error mitigation (ML-QEM) has emerged as a promising approach for improving the performance of noisy quantum algorithms. However, existing ML-QEM methods often have restricted applicability to variational circuits and rely on inaccessible noiseless training data. In this work, we propose a practical ML-QEM protocol tailored to variational quantum algorithms, which generates training data by simulating (near-)Clifford circuits. This data is used for model selection and training, producing a mitigation model that can correct variational circuits with arbitrary parameters and transfer across different target Hamiltonians of similar structure. We benchmark the proposed method on the Variational Quantum Eigensolver (VQE) task for the Sherrington-Kirkpatrick Hamiltonian of up to n=12n=12n=12 qubits under various noise models, analyzing its effect on trainability and comparing its performance against standard Zero-Noise Extrapolation (ZNE). The results demonstrate consistent several-fold error suppression across all tested settings and superior performance over ZNE in the high-noise regime, providing evidence for the applicability of the proposed protocol to present-day NISQ processors.
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.
Abstract Variational quantum algorithms (VQAs) are limited by device noise, and error mitigation is the standard remedy on near-term hardware. Choosing a mitigation method for deployment requires comparing its true cost, yet the usual comparison hides it: it fixes the number of shots per circuit execution, while the methods differ sharply in how many executions they need per estimate. Zero-noise extrapolation (ZNE) needs three to five, Clifford data regression (CDR) more than thirty per target circuit, and a trained neural model exactly one; ZNE's accuracy further rests on a noise-scaling assumption that real devices violate. Inside a variational loop that evaluates thousands of parameter settings, it is this execution multiplicity, not the accuracy at unlimited shots, that decides which method is affordable. Our premise is that the comparison that matters is at equal total shot budget per evaluation point. We present budget-conditioned neural error mitigation (NEM): one network corrects a single shot-limited measurement from circuit and device features, conditioned on the shot-noise scale so that one model serves every budget. We evaluate every method—including each ZNE fold point and each CDR training circuit—under a protocol that spends the same total shots per evaluation point. Under it, NEM is the most accurate method at every budget under coherent miscalibration (reducing error by 72–81%), and the only method that beats unmitigated execution at budgets up to 2^14 shots under stochastic device noise (11–20%); under miscalibration CDR is also positive but overtakes NEM only at 2^16 shots, at 32x the per-evaluation cost. We further characterize when extrapolation fails: under control-offset miscalibration, folding ZNE returns a worse answer than no mitigation on 38–63% of instances in simulation, and on up to 88% on a real IBM Heron processor. On that processor, fine-tuning the model on a small on-device set—120 circuits with classically computed labels—narrows the simulation-to-hardware gap, recovering a 39% error reduction on held-out device instances from the same calibration epoch. Improvements hold from 4 to 20 qubits. Budget-aware accounting, we argue, is the comparison that matters for deployment.