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arXiv:2605.08251

A finite-shot mean-squared-error boundary for Richardson zero-noise extrapolation

Vicenzo Scavino Alfaro

quant-phSubmitted May 7, 2026• Announced May 8, 2026• 0 citations• Revised Oct 15, 2026 (v2)
v1 · May 7, 2026v2 · Oct 15, 2026
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Abstract

Zero-noise extrapolation (ZNE) reduces the bias of noisy quantum-circuit measurements, but Richardson coefficients and shot splitting increase their statistical error. This paper determines the noise strength at which Richardson ZNE starts to lower the mean-squared error relative to the unmitigated estimator at a fixed shot budget $B$. The boundary is set by the balance between the leading squared-bias improvement and the leading excess-variance penalty: depending on the variance exponent of the measurement, it shrinks with $B$, becomes a budget threshold, or disappears. For fixed Richardson extrapolation with linear leading bias, the Richardson order and the shot allocation change only constants; the boundary exponent is set by the measured observable and protocol. Deterministic stabilizer measurements and nonzero-variance Hamiltonian or variational-energy measurements have boundaries that shrink as $B^{-1}$ and $B^{-1/2}$, and for finite-outcome shot measurements these two classes are the only ones. Qiskit Aer simulations of Greenberger-Horne-Zeilinger (GHZ) and quantum approximate optimization algorithm (QAOA) circuits reproduce this separation, and on IBM Quantum hardware ZNE lowers the mean-squared error at equal budget in 75 of 101 instances.

Comments
26 pages, 9 figures, 10 tables. v2: revised and extended version with a new title; restructured theory (general local theorem, fixed-Richardson derivation, observable classes), full simulation details, equal-budget analysis of all IBM hardware runs, and a supercritical synthetic control. Data and code: Zenodo, DOI 10.5281/zenodo.23405717

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Zero-noise extrapolation (ZNE) reduces the bias of noisy quantum-circuit measurements, but Richardson coefficients and shot splitting increase their statistical error. This paper determines the noise strength at which Richardson ZNE starts to lower the mean-squared error relative to the unmitigated estimator at a fixed shot budget BBB. The boundary is set by the balance between the leading squared-bias improvement and the leading excess-variance penalty: depending on the variance exponent of the measurement, it shrinks with BBB, becomes a budget threshold, or disappears. For fixed Richardson extrapolation with linear leading bias, the Richardson order and the shot allocation change only constants; the boundary exponent is set by the measured observable and protocol. Deterministic stabilizer measurements and nonzero-variance Hamiltonian or variational-energy measurements have boundaries that shrink as B−1B^{-1}B−1 and B−1/2B^{-1/2}B−1/2, and for finite-outcome shot measurements these two classes are the only ones. Qiskit Aer simulations of Greenberger-Horne-Zeilinger (GHZ) and quantum approximate optimization algorithm (QAOA) circuits reproduce this separation, and on IBM Quantum hardware ZNE lowers the mean-squared error at equal budget in 75 of 101 instances.
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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.
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.
Error mitigation is expected to turn noisy quantum measurements into more useful estimates. We show that it can instead amplify error. Under control-offset miscalibration, the evaluated unconstrained zero-noise extrapolation (ZNE) estimators are worse than no mitigation on 38-63% of simulated instances and produce extreme tail errors. A small IBM Heron study finds worsening on 80-88% of 18 instances across two depths, with mean error 3.0-4.1 times the raw error. In simulation, the median changes little, so median-only monitoring misses the failures. We then compare methods at the same online shot budget per evaluation and report offline training cost separately. After that cost is amortized, a budget-conditioned neural corrector occupies the low-budget end of the simulated accuracy-cost frontier and falls back to the raw estimate when an ensemble disagrees; its hardware transfer remains unsuccessful. Finally, we treat provider-reported calibration as an input that is not bound to the execution-time device state. A white-box projected-gradient stress test on this metadata increases the corrector's error 8.1 times within the feature ranges used for training and evades the disagreement monitor on half of the worsened cases. Together, the results connect budget-aware evaluation, tail risk, and calibration integrity in near-term quantum learning pipelines.
Zero-noise extrapolation (ZNE) is a widely used quantum error mitigation technique that artificially amplifies circuit noise and then extrapolates the results to the noise-free circuit. A common ZNE approach is Richardson extrapolation, which relies on polynomial interpolation. Despite its simplicity, efficient implementations of Richardson extrapolation face several challenges, including approximation errors from the non-polynomial behavior of noise channels, overfitting due to polynomial interpolation, and exponentially amplified measurement noise. This paper provides a comprehensive analysis of these challenges, presenting bias and variance bounds that quantify approximation errors. Additionally, for any precision ε , our results offer an estimate of the necessary sample complexity. We further extend the analysis to polynomial least squares-based extrapolation, which mitigates measurement noise and avoids overfitting. Finally, we propose a strategy for simultaneously mitigating circuit and algorithmic errors in the Trotter-Suzuki algorithm by jointly scaling the time step size and the noise level. This strategy provides a practical tool to enhance the reliability of near-term quantum computations. We support our theoretical findings with numerical experiments.
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.