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arXiv:2610.33386

Decision-aware validation of zero-noise extrapolation in QAOA optimization landscapes

Vicenzo Scavino Alfaro

quant-phSubmitted Oct 24, 2026• Announced Oct 27, 2026• 0 citations
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Abstract

Zero-noise extrapolation (ZNE) is usually validated by the accuracy of mitigated observables, yet variational quantum optimization uses those observables to select parameters. We ask whether, and by how much, an accuracy gain from ZNE coincides with a worse decision. In depth-one simulations of the quantum approximate optimization algorithm (QAOA) for MaxCut on four to six qubits, each sampled $120\times120$ landscape yields both its mean-squared error (MSE) and the ideal regret of its selected grid minimum, comparing Richardson orders one to three with unmitigated estimation at the same total shot budget per grid point. In an exploratory screen of 675 conditions, whose per-point random seeds collide, a familywise (Holm) analysis retains only four of 540 replication conditions as showing both directional MSE improvement and regret increase. A frozen confirmation with 100 fresh paired repetitions per condition reproduces all four: median MSE improvements of 7.65-73.09% accompany median normalized regret increases of only 0.0031-0.0073. We distinguish a bias route, in which residual bias reorders the mean landscape, from a variance route, in which Richardson weights amplify finite-shot noise. Exact first and second moments on a bounded control grid find no bias-route reversal: mitigated mean landscapes preserve the ideal minima, and ZNE can more than halve expected MSE, yet finite-shot selections incur higher mean regret. Decision losses that survive strict testing are few and small in this benchmark, but reproducible and not revealed by aggregate MSE. We present a paired benchmark and recommend reporting decision regret alongside estimation error when mitigation is assessed for optimization. Results concern simulated exhaustive grids, not hardware or optimizer loops.

Comments
27 pages, 3 figures, 6 appendices. Code and data: https://doi.org/10.5281/zenodo.23607395

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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.
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.
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.
Reliable benchmarking of Quantum Error Mitigation (QEM) requires distinguishing genuine improvements from artefacts of the post-processing arithmetic. In this paper, we expose a failure mode in Richardson Zero-Noise Extrapolation (ZNE), a widely used technique routinely (and often implicitly) relied upon in benchmarks and experiments. When noise amplification operates beyond usable signals - a regime that is quickly reached on current hardware for non-trivial circuits - we show that the extrapolation no longer reflects the underlying physics, but collapses into a fixed rescaling of a single noisy measurement, producing a bogus apparent improvement that is independent of noise amplification. This poses a rarely considered threat to the validity of many empirical evaluations in quantum computing. Measurements on real hardware (IQM Euro-Q-Exa) confirm this collapse with ordinary folding alone: as circuit depth erodes the signal, the reported estimate decouples from the truth and overshoots the ideal by up to 21%. We further introduce a matched-cost "garbage-folding" negative control that carries no usable signal yet reports a larger apparent improvement than genuine folding - showing that the magnitude of an improvement is not evidence of its correctness - alongside a zero-cost check flagging the artefact from data a benchmark already holds. We distil both into a short reporting checklist for ZNE benchmarks.
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.
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.