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To tell whether quantum error mitigation (QEM) improved a circuit result, compare the mitigated and unmitigated estimates with the same justified reference for the same circuit and observable. Report the errors and their uncertainties, then account for the extra executions, shots, calibration, and time the method requires. A changed estimate is not, by itself, evidence of greater accuracy.
Define “improvement” before running the experiment
An error-mitigation method improves a result only if it brings the estimate closer to a target that has been defined independently of the mitigation procedure. That target might be an ideal expectation value, a trusted probability, a reference energy, or another quantity appropriate to the task. The benchmark must represent the same circuit and observable being evaluated.
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For example, if the target expectation value is r, and the unmitigated and mitigated estimates are xraw and xmit, compare their absolute errors:
Raw error: |xraw − r|
Mitigated error: |xmit − r|
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The observed error reduction is the raw error minus the mitigated error. A positive difference means the mitigated point estimate is closer to the reference in that comparison; a negative difference means it is farther away. Neither observation alone tells you whether the difference is meaningful in the presence of statistical uncertainty.
Choose a defensible reference
When an exact or ideal result is available, state how it was obtained and what assumptions it relies on. If an exact reference is infeasible, select a justified proxy or benchmark and explain its limitations. A method’s internal consistency, agreement between repeated runs, or movement toward an expected-looking value is not a substitute for a reference. Without a justified target, you can report how estimates changed, but you cannot establish that the change made them more accurate.
State what quantity is being judged
Be explicit about the observable and the result being compared: for example, an expectation value, a probability, or an energy. An estimate may shift closer to the reference for one observable and farther away for another. Accuracy is therefore a claim about a specified quantity under specified conditions, not a general property that attaches to a method after one favorable result.
Run a matched comparison
The cleanest comparison changes the mitigation treatment while keeping the task fixed. Use the same target circuit, observable, compilation assumptions, backend, and run conditions for the raw and mitigated evaluations. If a difference in compilation or execution conditions cannot be avoided, record it; otherwise a difference attributed to mitigation may instead reflect a changed task or hardware condition.
- Record the target. Identify the circuit, observable, reference value or proxy, and how the reference was established.
- Record the execution context. Name the device or simulator, relevant device and software configuration, compilation assumptions, and execution date. Hardware behavior can change, so the date helps define the conditions to which the result applies.
- Run the unmitigated baseline. Save the raw estimate, shot count, and any repetitions or randomizations used.
- Run the mitigated procedure. Apply the chosen method to the same task and preserve the data needed to reproduce its estimate, including its additional settings and executions.
- Compare against the same reference. Calculate the raw and mitigated errors, report uncertainty for both estimates, and show the observed difference.
- Account for the resources. Report the additional executions, shots, calibration or characterization work, and relevant time or compute cost.
- Repeat across the intended scope. Test suitable circuits, observables, and noise conditions, and include cases where the method did not help.
Do not interpret “matched” as requiring identical shot counts if the mitigation algorithm necessarily uses a different allocation. Instead, disclose the allocation and distinguish two questions: whether the method improved the estimate under the runs performed, and whether it improved accuracy at a comparable total resource budget.
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Report uncertainty, not just point estimates
Finite sampling means an estimate has uncertainty. Give the shot count and uncertainty for the raw and mitigated results, identify how each uncertainty was calculated, and state how the uncertainty in the reference was handled. Where the reference itself is estimated rather than exact, its uncertainty matters to the comparison.
For stochastic methods or randomized circuit ensembles, report variation across randomizations or repetitions in addition to the uncertainty within an individual estimate. These are different sources of variation: a small sampling error in one run does not show that the method behaves consistently across randomized instances or repeated executions.
Show the uncertainty in the comparison
The decision concerns the difference between errors, not merely whether one estimate has a visually smaller error bar. Describe how uncertainty propagates through the estimator, any averaging, and any fitting procedure. If the observed reduction is small relative to the uncertainty in that reduction, say that the comparison is inconclusive rather than labeling the method a success.
A result can be closer to the reference in one run and still fail to establish a reliable improvement. Conversely, an error bar’s size alone does not determine whether a method improves accuracy: the relevant questions are how the point estimates compare with the target, how uncertain that comparison is, and what resources were used to obtain it.
Ratios need extra care near zero
An error ratio or improvement factor can be useful, but define it explicitly. For example, one possible ratio is raw absolute error divided by mitigated absolute error; it is not a universal standard. Ratios become unstable or difficult to interpret when the baseline error is near zero, and they can look dramatic when a denominator is very small. In that situation, show the absolute errors and their uncertainties alongside the ratio, and explain the rule used to interpret it.
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The 2023 IEEE Transactions on Quantum Engineering study uses an empirically motivated, resource-normalized “improvement factor” for QEM. Attribute that metric to its authors rather than presenting it as a universal pass criterion. The available evidence does not establish a generally applicable minimum improvement, confidence threshold, or success percentage. Define the decision rule for the experiment instead of borrowing an unsupported cutoff.
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A method that reduces error may require additional work. Report the cost that is relevant to the experiment: extra circuit executions and shots, calibration or characterization effort, and time or compute cost where applicable. If one method uses substantially more resources than the raw baseline, its result should not be presented as a cost-free improvement.
Where feasible, compare methods at matched total resource budgets. If that is impractical, state the normalization used and report both the accuracy comparison and the resource difference. A resource-normalized metric can help summarize the trade-off, but it does not replace the underlying error, uncertainty, and cost figures. Readers need enough information to decide whether the extra accuracy, if established, is worth the extra resources in their setting.
Interpret the evidence by method
Different QEM methods make different assumptions and incur different costs. Use the same reference, matched-task discipline, uncertainty reporting, and resource accounting when comparing candidates. A method should not win simply because it was given a more favorable benchmark, more executions, or a less demanding success criterion.
| Method | What to document | How to interpret the evidence |
|---|---|---|
| Zero-noise extrapolation (ZNE) | Effective noise strengths, amplification factors, measured points, fitting or extrapolation model, and uncertainty propagation. | The extrapolated result depends on the amplification and fit choices; it is not guaranteed to improve the raw estimate. |
| Probabilistic error cancellation (PEC) | Benchmark, reference, shot and execution costs, uncertainty, and any method-specific settings used. | Compare it on the same terms as other methods. Published platform evidence does not establish that PEC wins on every device or problem. |
| Randomized compiling combined with ZNE | Randomization procedure, ZNE amplification and fitting choices, repetitions, uncertainty, and resources. | Published VQE evidence described below comes from numerical simulation for small molecules and studied coherent-noise conditions. |
Zero-noise extrapolation: disclose the path to the zero-noise estimate
ZNE estimates results at multiple effective noise strengths and extrapolates toward the zero-noise limit. The estimated limit depends on both the noise-amplification factors and the model or fit applied to the measured points. Report the factors, the points, the fitting choices, and the uncertainty associated with that extrapolation; a final extrapolated number alone is not enough to assess the claim.
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Koenig, Reinecke, Hahn, and Wellens emphasize this issue in their Physical Review A paper, published October 24, 2024: “For accurate extrapolation, it is essential to know the exact factors of the noise amplification.” Their paper presents an inverted-circuit procedure that appends the inverse circuit and measures the probability of returning to the initial state to estimate error strength. That is evidence about the studied procedure, not proof that amplification factors are known accurately in every ZNE experiment.
The original noise level also matters. The paper qualifies ZNE performance when the original noise becomes too large. Do not turn a result from the studied method and conditions into a claim that extrapolation always helps. A fit may produce a value closer to the reference in one setting while producing no gain or a worse estimate in another.
Probabilistic error cancellation: evaluate accuracy and overhead together
PEC appears alongside ZNE in the 2023 IEEE platform study. Its performance should be assessed against the same target and uncertainty standards, with the added execution and sampling costs made visible. The study’s abstract supports a conditional conclusion: on average, mitigation was more beneficial than no mitigation even when normalized by additional resources, while performance depended on the underlying computer. That is not evidence that PEC—or any single method—will improve every circuit on every backend.
Randomized compiling plus ZNE: preserve the simulation scope
Kurita and colleagues’ Quantum paper, published November 20, 2023, proposes combining randomized compiling (RC) with ZNE for the variational quantum eigensolver (VQE). Its stated motivation is that RC changes coherent errors into stochastic Pauli errors, which can make extrapolation more tractable. The paper’s reported evidence is numerical simulation for small-molecule VQE, not a general hardware guarantee.
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In those simulations, the authors reported up to two orders of magnitude lower energy errors induced by the studied types of coherent noise. Retain every part of that qualification when citing the figure: it is an upper reported reduction for the simulated VQE cases and coherent-noise conditions studied, not a typical gain, a hardware benchmark, or a forecast for arbitrary circuits.
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Test whether the result is robust
One circuit and one observable can establish only a narrow result. To support a broader claim, repeat the evaluation across circuits, observables, and noise conditions relevant to the intended use. Report cases with no improvement or degradation as well as successful cases; omitting failures makes it impossible to judge the method’s range.
Hardware or backend dependence is material. The IEEE study, published August 15, 2023, tested ZNE and PEC on two benchmark problems across IBM, IonQ, Rigetti, and noisy simulators. Its findings underscore platform-dependent performance. This is a scoped study, not an exhaustive or current ranking of providers, and it does not establish that a method’s performance transfers unchanged to another computer or workload.
- Keep the reference and observable definition consistent across the comparisons.
- Vary circuits or noise conditions only when that variation is part of the question, and record what changed.
- Report the number of repetitions or randomized instances and how their results were aggregated.
- Include both successful and unsuccessful cases, with the uncertainty and resources for each.
- Limit conclusions to the tested circuits, observables, backend, and conditions.
Common evaluation mistakes
- Reporting only the mitigated value: without the raw baseline and reference, readers cannot tell whether the value became more accurate.
- Treating a shift as proof: an estimate can move substantially without moving closer to the target.
- Calling a single favorable run a win: if uncertainty makes the difference inconclusive, state that rather than implying a dependable improvement.
- Showing an error bar without its method: include shot counts, uncertainty calculations, fitting details, and repeated-run or randomization variation where relevant.
- Hiding overhead: disclose added executions, shots, calibration or characterization, and relevant time or compute cost.
- Generalizing beyond the evidence: simulation results, one backend, or a small benchmark set do not demonstrate universal hardware performance.
- Assuming the extrapolation model is correct: ZNE conclusions depend on noise amplification and fitting choices, especially when the original noise is high.
- Using a ratio without the underlying errors: a near-zero baseline or denominator can make a ratio misleading; show absolute errors and uncertainty too.
What a complete evaluation report should contain
A reader should be able to reconstruct what was compared and understand why the result counts—or does not count—as an improvement. A concise report can use this checklist:
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- Reference: value or proxy, its source, and its limitations.
- Conditions: device or simulator, relevant software and device configuration, compilation assumptions, noise regime, and execution date.
- Baseline and treatment: raw and mitigated estimates, shot counts, repetitions, and method settings.
- Method details: for ZNE, amplification factors, measured noise points, and fitting or extrapolation procedure; for randomized methods, ensemble and repetition details.
- Uncertainty: estimator, uncertainty calculation, propagation through fitting or averaging, and reference uncertainty if applicable.
- Accuracy comparison: absolute raw and mitigated errors, their observed difference, and any defined ratio or improvement factor.
- Resource accounting: additional executions, shots, calibration or characterization, and relevant time or compute cost, plus the chosen resource normalization.
- Scope: circuits, observables, backends, and conditions tested, including cases without improvement.
- Conclusion: whether the evidence supports improvement under those conditions, whether uncertainty leaves the result inconclusive, and what cannot be generalized.
The conclusion should follow from the comparison, not from the method’s name or its intended purpose. If the mitigated estimate is closer to a justified reference and that reduction is supported by the uncertainty analysis, you can report an improvement for the tested scope. If it is not closer, or if uncertainty prevents a clear distinction, report that outcome just as plainly.
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