Error mitigation changes how observations are used to estimate a quantity. Verification asks whether the resulting claim survives appropriate checks. These are connected tasks: a correction can reduce modeled bias while increasing variance or becoming sensitive to a wrong calibration. A corrected number closer to an ideal value is not, by itself, evidence that the estimate improved.
Begin with an explicit target. The lab's raw Bell score measures correlations after its simulated readout process. The prepared state's Bell overlap is a different quantity. At p=0.02 and q=0.01, the analytic values are 0.955894 for the expected observed score and 0.985 for the pre-readout overlap. The difference is produced by the declared model, not by finite sampling. Increasing shots estimates the first quantity more precisely; it does not make that quantity become the second.
Work through a correction that can be checked
Suppose each measured bit independently flips with exactly known probability q. Each bit's sign is attenuated by 1−2q, so a two-bit correlation is attenuated by
c = (1 - 2q)^2.In this restricted symmetric model, divide the observed correlations by c. The score and its conditional uncertainty transform as
S_corrected = 1/4 + (S_raw - 1/4)/c
w_corrected = w_raw/c.The offset 1/4 is essential. Dividing the entire score by c is wrong. The formulas assume the same attenuation in all three bases and a valid, known calibration. More general measurement mitigation uses a response matrix and must confront its conditioning and calibration cost. Bravyi and colleagues' measurement-mitigation study.
For the baseline, c=0.9604. The analytic expected raw score 0.955894 transforms to 0.985. At 5,000 shots per basis, the approved raw half-width 0.038923 becomes 0.040528 after correction. This corrected interval fails the 0.04 precision condition, even though the raw interval satisfies it. The committed Aer capture's raw estimate is 0.956100; applying the same known-q calculation gives approximately 0.985214. These are different inputs to the same transformation: one is a model expectation, the other a captured sample estimate.
Do not relabel the lab's raw-score acceptance as a mitigated-fidelity acceptance. A new target quantity needs a declared estimation and decision contract.
Pay attention to variance and calibration uncertainty
Scaling a random quantity by 1/c scales its variance by 1/c². At q=0.01, that multiplier is about 1.08417 relative to the same raw observations. At q=0.10, c=0.64 and the multiplier is 2.44141. The correction has not created additional observations. It has amplified the signal and its sampling fluctuation.
These multipliers do not describe the entire cost relative to perfect readout: the raw noisy experiment already has a different variance, and acquiring calibration data costs shots too. As q approaches 0.5, the inverse becomes unstable because measured bits carry little information about their inputs.
When q is estimated, treating it as exact understates uncertainty. Allocate uncertainty to the calibration and science samples, then propagate both through the correction. A calibration collected before drift may describe the wrong response. Basis-dependent or asymmetric errors also invalidate the single-c formula. Preserve calibration identifiers, acquisition conditions, raw outcomes, and the transformation so another reader can test the assumptions.
An inverse estimator can fall outside a physical range because of sampling. Clipping may be a useful explicitly defined estimator, but it changes its bias and cannot be presented as unchanged unbiased inversion. Retain the unmodified calculation for diagnosis.
Verification should challenge the correction
Use cases with known answers to test signs, normalization, basis rotation, and the correction itself. The Bell example is small enough to compare analytical probabilities with a density-matrix simulator. Agreement tests the implementation against the chosen model; it does not establish that the model describes a real processor.
Then challenge the model. Vary calibration age, use separate validation data, inspect residuals by basis, and compare corrected and raw results with their respective uncertainty. A high corrected estimate accompanied by disagreement across independent diagnostics is a reason to investigate. Reusing the data that selected the correction as its only validation can hide overfitting.
Other mitigation methods also need explicit assumptions. An extrapolation toward lower noise depends on the extrapolation model; randomized inverse methods can incur large sampling overhead. Neither justifies replacing a measured uncertainty with an attractive central value. Foundational mitigation methods. The general design question remains: which bias is addressed, under what model, and at what statistical cost?
Exercise and worked answer
In a constructed arithmetic exercise, take q=0.10, a raw score of 0.7204, and raw half-width 0.0389232. Compute the corrected center and half-width. Would the original precision requirement pass? What additional uncertainty is omitted if q came from a finite calibration experiment?
Worked answer: c=0.64. The corrected center is 0.25+(0.7204−0.25)/0.64=0.985. The half-width is 0.0389232/0.64=0.0608175, so the 0.04 requirement fails. The variance multiplier is 1/0.64²=2.44140625. Calibration uncertainty, possible drift, and model mismatch remain outside that conditional interval. The correct next step is a justified acquisition and verification plan, not rounding or clipping the width. Shots, Uncertainty, and Stopping Rules supplies the raw statistical contract.