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arxiv: 2307.05302 · v3 · submitted 2023-07-11 · 🪐 quant-ph

Robust design under uncertainty in quantum error mitigation

Maksym Prodius , Piotr Czarnik , Michael McKerns , Andrew T. Sornborger , Lukasz Cincio This is my paper

Pith reviewed 2026-05-24 07:17 UTC · model grok-4.3

classification 🪐 quant-ph
keywords error mitigationzero noise extrapolationClifford data regressionuncertainty quantificationrobust optimizationquantum computingXY model
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The pith

Strategic sampling of mitigation outcomes quantifies uncertainty and enables robust hyperparameter optimization in quantum error mitigation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops methods to estimate the uncertainty in error-mitigated quantum observables by strategically sampling different mitigation results rather than relying on single runs. These uncertainty estimates then guide the optimization of key design choices, such as which noise levels to extrapolate from and how to allocate measurement shots in zero-noise extrapolation, or which circuits to use for training in Clifford data regression. The optimizations aim to make the mitigated results more accurate and less sensitive to the specific noise present in the device. A reader would care because current error mitigation techniques suffer from uncertainty due to limited quantum measurements, limiting their reliability for practical computations on near-term hardware.

Core claim

By sampling a statistical distribution of error mitigation outcomes, unbiased estimates of the uncertainty and error in mitigated observables can be obtained. This sampling approach is extended to perform optimization of mitigation hyperparameters, including noise levels and shot allocation for zero noise extrapolation and the distribution of training circuits for Clifford data regression, using surrogate-based methods to keep the overhead manageable. The resulting designs show improved performance and transferability across similar circuits when applied to simulations of the XY model ground state under depolarizing and IBM Toronto noise.

What carries the argument

Strategic sampling of error mitigation outcomes to build uncertainty distributions, paired with surrogate-based optimization of mitigation hyperparameters.

If this is right

  • Better choices of noise levels and shot numbers lead to lower error in zero-noise extrapolation results for the XY model.
  • Optimized training circuit sets improve the accuracy of Clifford data regression.
  • Learned hyperparameters from one circuit can be transferred to similar ones without re-optimization.
  • The uncertainty quantification applies generally to any classical post-processing error mitigation method.
  • Surrogate optimization allows efficient hyperparameter tuning despite the need for multiple samples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The methods could help in designing mitigation protocols that adapt to changing noise conditions during a computation.
  • Similar sampling ideas might apply to other quantum tasks like variational algorithms where post-processing is used.
  • If the overhead is small, these robust designs could become standard for reducing effective error rates in families of circuits.
  • The transferability suggests potential for pre-computing optimal settings across related problem instances.

Load-bearing premise

The extra sampling required to estimate uncertainties and run the hyperparameter optimizations remains computationally affordable and does not erase the gains from mitigation.

What would settle it

An experiment where the uncertainty predicted by the sampling method for a mitigated observable differs significantly from the actual variation observed across hundreds of independent full runs of the mitigation protocol on the same circuit.

Figures

Figures reproduced from arXiv: 2307.05302 by Andrew T. Sornborger, Lukasz Cincio, Maksym Prodius, Michael McKerns, Piotr Czarnik.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: This result demonstrates a strong dependence of the CDR performance on the training data distribution. The resulting parameters correspond to the strongest [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

Error mitigation techniques are crucial to achieving near-term quantum advantage. Classical post-processing of quantum computation outcomes is a popular approach for error mitigation, which includes methods such as Zero Noise Extrapolation, Virtual Distillation, and learning-based error mitigation. However, these techniques have limitations due to the propagation of uncertainty resulting from the finite shot number of a quantum measurement. In this work, we introduce general and unbiased methods for quantifying the uncertainty and error of error-mitigated observables based on the strategic sampling of error mitigation outcomes. We then extend our approach to demonstrate the optimization of performance and robustness of error mitigation under uncertainty. To illustrate our methods, we apply them to Zero Noise Extrapolation and Clifford Date Regression in the ground state of the XY model simulated using depolarizing and IBM Toronto noise models, respectively. In particular, we optimize the choice of noise levels and the allocation of shots for Zero Noise Extrapolation and the distribution of the training circuits for Clifford Data Regression. While our methods are readily applicable to any post-processing-based error mitigation approach, in practice they must not be prohibitively expensive -- even though they perform optimizations of the error mitigation hyperparameters requiring sampling of a statistical distribution of error mitigation outcomes. By leveraging surrogate-based optimization, we show that our methods can efficiently perform optimal design for a Zero Noise Extrapolation implementation. We then further demonstrate the transferability of learned Zero Noise Extrapolation hyperparameters to other similar circuits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to introduce general unbiased sampling-based methods for quantifying uncertainty and error in post-processing error-mitigated observables, then extends the framework to surrogate-based optimization of mitigation hyperparameters (noise levels, shot allocation, training-circuit distribution) for robustness under finite-shot uncertainty. Demonstrations apply the methods to zero-noise extrapolation (ZNE) on the XY-model ground state with depolarizing noise and to Clifford data regression (CDR) with IBM Toronto noise, including transferability of learned ZNE hyperparameters.

Significance. If the sampling-based estimators are unbiased and the surrogate overhead remains sub-dominant, the work supplies a practical route to robust hyperparameter design for error mitigation, addressing a key barrier to reliable near-term quantum computation. The explicit use of surrogate optimization to control the cost of uncertainty-aware tuning is a methodological strength that could generalize beyond the two techniques shown.

major comments (2)
  1. [§4, §5] §4 (ZNE results) and §5 (CDR results): the demonstrations report improved mitigated values and reduced variance after optimization but supply no table or figure comparing total shot expenditure (including all surrogate samples) against the unoptimized baseline or against the raw mitigation gain; without these counts the central practicality claim cannot be verified.
  2. [§3.2] §3.2 (surrogate optimization): the claim that the additional sampling 'must not be prohibitively expensive' is stated as a necessary condition, yet no analytic or numerical bound is derived on the regime where the surrogate overhead remains smaller than the mitigation benefit; the XY-model examples therefore leave the load-bearing affordability assumption untested.
minor comments (2)
  1. [§3] Notation for the sampled error-mitigation outcomes (e.g., the random variable whose expectation yields the unbiased estimator) is introduced without an explicit equation number in the methods section, making it harder to trace the unbiasedness argument.
  2. [Figures 3-5] Figure captions for the hyperparameter landscapes do not state the number of independent surrogate runs used to generate the plotted surfaces, which affects reproducibility of the optimization results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [§4, §5] §4 (ZNE results) and §5 (CDR results): the demonstrations report improved mitigated values and reduced variance after optimization but supply no table or figure comparing total shot expenditure (including all surrogate samples) against the unoptimized baseline or against the raw mitigation gain; without these counts the central practicality claim cannot be verified.

    Authors: We agree that explicit accounting of total shot expenditure, including surrogate overhead, is required to substantiate practicality. In the revised manuscript we will add tables in §§4 and 5 that report the aggregate shot counts for the full surrogate-optimization workflow, the unoptimized mitigation baseline, and the raw (unmitigated) case, allowing direct comparison of net resource cost versus mitigation gain. revision: yes

  2. Referee: [§3.2] §3.2 (surrogate optimization): the claim that the additional sampling 'must not be prohibitively expensive' is stated as a necessary condition, yet no analytic or numerical bound is derived on the regime where the surrogate overhead remains smaller than the mitigation benefit; the XY-model examples therefore leave the load-bearing affordability assumption untested.

    Authors: We acknowledge the absence of an explicit bound. While a general analytic bound is difficult to derive because overhead depends on circuit structure and noise model, we will revise §3.2 to include a quantitative numerical comparison drawn from the XY-model demonstrations that shows the surrogate overhead relative to the observed mitigation improvement. This will make the affordability assumption testable for the reported cases. revision: partial

Circularity Check

0 steps flagged

No significant circularity; new sampling-based uncertainty methods and surrogate optimization are independent of fitted inputs or self-citation chains.

full rationale

The paper introduces general methods for quantifying uncertainty in error-mitigated observables via strategic sampling of outcomes, then extends to hyperparameter optimization under uncertainty using surrogate-based optimization. These steps rely on standard statistical sampling applied to existing error mitigation techniques (ZNE, CDR) rather than redefining or fitting quantities that loop back to the outputs. The abstract and described claims contain no self-definitional equations, no fitted parameters renamed as predictions, and no load-bearing self-citations or uniqueness theorems imported from prior author work. The affordability condition is explicitly flagged as a practical requirement but is not part of any derivation that reduces the central results to inputs by construction. The approach remains self-contained, with demonstrations on the XY model serving as illustrations rather than circular validations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated. The optimization targets (noise levels, shot allocations, training-circuit distributions) function as tunable quantities whose values are chosen to minimize uncertainty, but their status as fitted versus externally fixed cannot be determined from the given text.

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Reference graph

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