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The only defensible test is a matched accuracy comparison. Run the same circuit and observable with and without mitigation, compare both estimates with an explicitly defined ideal or trusted reference, quantify uncertainty, and include every resource the mitigation consumed. A changed expectation value, a smaller-looking error bar, or a result that “looks more physical” is not evidence of improvement by itself.
Quantum error mitigation (QEM) can improve estimates, but its benefit depends on the mitigation method, circuit, noise regime, hardware and resource budget. There is no universal percentage gain or pass/fail threshold. The procedure below gives you a reproducible decision rule for your particular experiment.
Define what “better” means before running the experiment
Choose the quantity you are trying to estimate: an expectation value, probability, energy, correlation, or another scalar derived from the circuit. Then define the value against which accuracy will be judged. The reference can be an exact calculation for a small circuit, a high-precision classical simulation, a trusted experiment, or a benchmark value with a documented uncertainty. State how it was obtained and what limitations it has.
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Use an explicit error metric
For a scalar estimate x and reference r, report absolute error:
E = |x − r|
For quantities where a relative comparison is meaningful, also report relative error:
Erel = |x − r| / |r|
Relative error is unstable when the reference is zero or close to zero. In that case, use absolute error, a problem-specific scale, or a confidence interval around the difference. For vector outputs, specify the norm or distributional distance instead of selecting whichever component improved.
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Define the decision rule
Call mitigation an improvement only if the mitigated estimate has lower error than the unmitigated estimate under the same stated conditions and the difference is supported by the uncertainty. A useful primary quantity is
ΔE = Eraw − Emitigated
Positive ΔE favors mitigation. Report its uncertainty or confidence interval. If the interval includes zero, describe the result as inconclusive rather than successful. If the raw error is already near zero, a ratio such as Eraw/Emitigated can become misleading; show the two errors and their absolute difference first.
Keep the comparison matched
The raw and mitigated estimates must represent the same computational task. Any difference in the circuit, observable, compilation, backend, or execution window can masquerade as a mitigation effect.
| Element to hold fixed | What to record | Why it matters |
|---|---|---|
| Target circuit | Gate sequence, qubit mapping, depth, inserted measurements and any circuit transformations | A different circuit has a different error channel and is not a controlled comparison. |
| Observable or output | Pauli term, energy functional, probability, bit string or distribution metric | Mitigation can help one observable while harming another. |
| Compilation | Compiler version, optimization settings, basis gates and transpiled circuit | Compilation changes gate count, duration and noise exposure. |
| Backend | Device or simulator, qubit selection, calibration identifier and noise model | QEM performance is hardware- and noise-model-dependent. |
| Execution conditions | Date, shot count, batching, reset and measurement settings | Calibrations and drift can change results between runs. |
Record the software and device configuration and the execution date. If the backend recalibrates between raw and mitigated runs, say so and treat the result as less tightly controlled.
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- Specify the target and reference. Write down the observable, the ideal or trusted value, how that value was obtained, and its own uncertainty or approximation error.
- Freeze the task. Fix the circuit, measurement definition, compilation assumptions, backend, noise model, and relevant run settings. Save the exact circuit representation used for both conditions.
- Collect the unmitigated baseline. Use a predeclared shot budget and repeat structure. Store raw counts or samples, not only the final mean.
- Run the mitigation protocol. Document all extra circuits, noise-amplification factors, randomizations, calibration or characterization jobs, fit model and estimator. Keep the target observable identical.
- Estimate uncertainty. Propagate shot noise and, where applicable, variation across randomized circuits, repetitions, calibration data and fitted parameters.
- Compute errors against the reference. Report
Eraw,EmitigatedandΔE, with intervals or standard errors. Do not infer accuracy from the mitigated value alone. - Normalize for resources. Count circuit executions and shots, plus calibration, characterization, queue or wall-clock time and classical fitting cost when relevant.
- Repeat across the intended scope. Test representative depths, observables and noise conditions. Include cases with no improvement or degradation.
Quantify uncertainty rather than reporting a single number
Sampling uncertainty is unavoidable. For a measured probability, the raw count data have binomial or multinomial variation; for an expectation value, the variance depends on the measured outcomes and shot count. Report the number of shots and the estimator used. A standard error or confidence interval should accompany every raw and mitigated estimate.
Mitigation often adds uncertainty beyond ordinary shot noise. A randomized method requires variation across random circuit instances. A fitted extrapolation requires uncertainty in each noise-scaled point and in the fit coefficients. Probabilistic estimators can have broad or heavy-tailed samples. Use repeated batches, bootstrap or another documented resampling method when an analytic propagation is not appropriate.
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For the comparison itself, estimate the uncertainty of the difference between errors or of the difference between the two estimates. If raw and mitigated data share shots, random seeds or calibration parameters, account for that covariance rather than treating them as independent. State the confidence level or interval construction.
What counts as a convincing result?
A convincing result has a predeclared reference and metric, a lower mitigated error, and an uncertainty interval for the improvement that is sufficiently separated from zero for the purpose of the experiment. The sources reviewed here establish no universal minimum improvement, confidence level or resource-normalized pass criterion. Choose and justify your rule instead of importing a made-up threshold.
Include the cost of mitigation
Mitigation can require many more circuits or shots than the baseline. Count:
- Additional circuit executions and total shots.
- Calibration, noise-characterization and verification jobs.
- Randomized circuit instances and repeats.
- Wall-clock or queue time when throughput matters.
- Classical fitting, sampling and storage work.
Compare methods at equal total resources when feasible. If equal-resource testing is impractical, state the normalization clearly—for example, error at a fixed number of shots or error achieved by a fixed wall-clock budget.
The 2023 IEEE Transactions on Quantum Engineering study introduced an empirically motivated, resource-normalized metric called an improvement factor. It is an example of putting overhead into the comparison, not a universal industry standard. Report the underlying errors and resource counts even if you also use that metric.
Evaluating zero-noise extrapolation
Zero-noise extrapolation (ZNE) measures the circuit at several effective noise strengths and fits those results to estimate the zero-noise limit. The extrapolated value is not an independent measurement: it depends on how noise was amplified, which points were measured and which fit or extrapolation model was chosen.
Document the noise-scaling experiment
- List every noise-amplification factor and how it was implemented.
- Show the measured value and uncertainty at each factor.
- State whether the scaling used gate folding, pulse or another transformation.
- Name the fit family, order, weighting, parameter constraints and extrapolation point.
- Report how many circuit repetitions and shots produced each point.
- Explain how fit uncertainty was propagated to the zero-noise estimate.
Do not present the extrapolated value without the measured points. A high-order fit can pass through noisy data while producing an unstable intercept; a low-order fit can be biased if the actual noise dependence is not close to its assumed form.
Why accurate noise factors matter
Koenig, Reinecke, Hahn and Wellens emphasize in their 2024 Physical Review A paper: “For accurate extrapolation, it is essential to know the exact factors of the noise amplification.” If the implemented factors differ from the factors used in the fit, the extrapolation is solving the wrong model.
The paper presents inverted-circuit estimation: append the inverse of the circuit and measure the probability of returning to the initial state to estimate error strength. This is a method-specific way to characterize the effective noise scale; it does not remove the need to report the measured factors and their uncertainty.
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Recognize ZNE failure modes
ZNE can overshoot, amplify variance or become unstable when the original noise is already large. The reported performance in the APS study is conditional on its method and tested regime, not a guarantee for every circuit. A mitigated value farther from the reference than the raw value is a failure for that observable, even if the fit appears smooth.
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Probabilistic error cancellation (PEC) is another method examined in the 2023 IEEE platform study. Evaluate it with the same baseline, reference, uncertainty and resource accounting as ZNE. In particular, report the number of sampled correction circuits and shots used to form the quasi-probabilistic estimator, along with any characterization data required to construct it.
PEC’s apparent accuracy at a small shot count can be deceptive if the estimator variance is high. Compare confidence intervals and error at a fixed total resource budget, not only central estimates obtained with different numbers of executions. Do not claim that PEC is superior on every device or problem: the IEEE results show that QEM performance depends on the underlying computer.
Randomized compiling combined with ZNE
Kurita and colleagues proposed combining randomized compiling (RC) with ZNE for variational quantum eigensolvers in a 2023 Quantum paper. The motivation is that RC transforms coherent errors into more stochastic Pauli-like errors, which can make extrapolation easier to model.
The reported evidence is numerical simulation for small-molecule VQE, not a general hardware benchmark. Under the studied coherent-noise conditions, the simulations reported up to two orders of magnitude lower energy errors. That figure applies to those simulated circuits and noise types; it does not establish the same reduction for arbitrary hardware, ansätze or observables.
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When testing RC+ZNE yourself, count every randomized compilation instance and repeat, hold the randomization budget fixed between methods, and include the variability across random seeds in the uncertainty. Compare energy error against the same reference energy and include the cost of generating and executing the randomized ensemble.
What cross-platform evidence does—and does not—show
The IEEE study tested ZNE and PEC on two benchmark problems using IBM, IonQ and Rigetti systems as well as noisy simulators. Its abstract reports that error mitigation was, on average, more beneficial than no mitigation even after normalizing by additional resources, while emphasizing that performance depends on the underlying computer.
“Our results show that error mitigation is, on average, more beneficial than no error mitigation—even when normalized by the additional resources used—but also emphasize that the performance of QEM depends on the underlying computer.”
This is evidence that a resource-aware, multi-platform evaluation is useful. It is not an exhaustive ranking of current providers, a guarantee for a new circuit, or a universal success rate. Generalize only to hardware, noise and benchmark families that your experiment actually covers.
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Compare candidate methods on the same six axes
| Axis | Questions to answer |
|---|---|
| Accuracy | How far is each estimate from the same reference, using the same observable and error metric? |
| Uncertainty | Are intervals reported, reproducible across repeats and inclusive of fit, randomization or estimator variance? |
| Resource overhead | How many extra circuits, shots, calibration jobs, seconds and classical operations are required? |
| Noise assumptions | Does the method rely on a characterized amplification factor, a noise model, stochasticity or another condition? |
| Robustness | Does it work across the depths, observables and noise strengths relevant to the application? |
| Backend dependence | Does the result persist across devices, qubit choices, calibration periods or simulators? |
Use a single primary metric and resource budget for the headline comparison, then publish secondary metrics so readers can see trade-offs. A method that has the smallest error with unlimited shots may be a poor choice when the real constraint is queue time or calibration stability.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Common evaluation mistakes and how to correct them
Reporting only the mitigated value
Problem: A reader cannot tell whether the raw estimate was already closer to the reference.
Correction: Show raw, mitigated and reference values, with uncertainty and the error difference.
Calling a smaller error bar an accuracy gain
Problem: An estimator can be more precise yet biased farther from the target.
Correction: Separate precision (uncertainty width) from accuracy (distance to the reference).
Using one favorable run
Problem: Sampling and drift can make a single run atypical.
Correction: Repeat batches or randomizations and report the distribution, not just the best outcome.
Ignoring the reference uncertainty
Problem: A supposedly exact target may itself be approximate or noisy.
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Correction: Include the reference’s numerical or experimental uncertainty in the interpretation.
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Hiding extrapolation choices
Problem: Different scaling factors or fit orders can produce different ZNE answers.
Correction: Publish factors, measured points, fit specification and sensitivity to reasonable alternative fits.
Comparing unequal budgets
Problem: Mitigation may appear better simply because it received many more shots.
Correction: Match total resources or label the unequal-budget comparison explicitly.
Generalizing simulation evidence to hardware
Problem: Simulated coherent noise may not reproduce crosstalk, drift, leakage or calibration behavior on a device.
Correction: State the tested noise model and restrict the claim to that scope.
A report format that makes the conclusion auditable
Include a compact table like this for every benchmark or circuit family:
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|---|---|
| Task | Circuit identifier, depth, qubits, compilation and observable |
| Reference | Value, method of generation and uncertainty |
| Backend | Device or simulator, noise model, qubits, calibration and execution date |
| Raw estimate | Value, shots, estimator and uncertainty |
| Mitigated estimate | Value, shots, estimator and uncertainty |
| Error comparison | Absolute errors, ΔE, interval and decision rule |
| Mitigation details | Scaling factors, randomizations, fit or characterization procedure |
| Resources | Total executions, shots, calibration work, time and classical cost |
| Scope | Circuits, observables and noise conditions for which the claim is made |
Then state the conclusion in one of three forms: improved (lower error with supported uncertainty), no demonstrated improvement (difference is inconclusive or statistically unsupported), or worsened (higher error under the stated budget). If the result changes with the resource budget or observable, report that conditional conclusion instead of forcing a single winner.
Quick Recap
Final checklist
- Is the target quantity and reference explicitly defined?
- Were raw and mitigated runs matched in circuit, observable, compilation, backend and conditions?
- Are shots, repeats, estimator details and uncertainty reported?
- Are absolute errors and their difference shown?
- Are near-zero baselines handled without a misleading ratio?
- Are extra executions, calibration, time and classical costs counted?
- For ZNE, are amplification factors, measured points and fit choices documented?
- For randomized methods, is across-randomization variation included?
- Were multiple relevant circuits and noise regimes tested, including failures?
- Does the conclusion stay within the tested hardware and noise scope?
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