An error budget becomes useful when it names an observable, an operating context, and a tolerable consequence. A single aggregate “error rate” rarely provides enough information to decide whether a quantum workload will succeed. Preparation faults, coherent control errors, relaxation, leakage, crosstalk, measurement errors, and sampling uncertainty affect results in different ways.
The teaching lab uses a deliberately restricted model so these distinctions can be calculated. Its preparation channel acts once on the two-qubit Bell state. Its measurement channel independently flips each recorded bit. Real devices can contain additional effects and correlations that this model omits.
Define the channel before interpreting its parameter
The implemented Aer channel has the form ρ′ = (1 − p)ρ + pI/4. Thus p = 0 means no added preparation channel, while p = 1 replaces the two-qubit state with the maximally mixed state. The parameter is a depolarizing mixing coefficient, not a per-gate hardware error rate or a generic probability that an arbitrary circuit fails.
For ideal Bell preparation followed by this channel, the prepared-state Bell overlap is F_prepared = 1 − 3p/4. At p = 0.02, it is 0.985. The factor of three quarters appears because the maximally mixed two-qubit state still has overlap one quarter with any particular pure state.
Noise APIs do not all use the same parameter convention. For example, a channel parameter can denote a mixing coefficient or the total probability of applying a nonidentity Pauli. Record the exact definition rather than transferring a slider value between simulators. The Aer depolarizing-channel reference specifies the convention used here.
Readout changes the observed quantity
Independent symmetric bit flips with probability q attenuate a two-bit parity correlation by (1 − 2q)². A flip on either one bit reverses parity; flips on both bits restore it. Combining the preparation and readout models gives the correlation magnitude t = (1 − p)(1 − 2q)².
The ideal signs remain positive for XX and ZZ and negative for YY. The expected raw score is consequently (1 + 3t)/4. At p = 0.02 and q = 0.01, t = 0.941192 and the expected score is 0.955894. This is lower than the prepared overlap 0.985 even with infinitely many shots.
That difference is systematic under the declared model. Sampling more often concentrates the estimator around the raw expectation; it does not make the expectation approach the prepared-state value. An explicit readout-correction model can target a different estimand, but introduces its own calibration assumptions and uncertainty.
Worked example: allocate distinct margins
Suppose the raw score must have a lower bound of at least 0.9. The baseline expectation leaves approximately 0.055894 between its center and that threshold. At 5,000 shots per basis, the lab's simultaneous half-width is approximately 0.038923. There is enough room for the captured estimate to satisfy the threshold, although the decision must use the actual estimate rather than the expectation.
Now increase q to 0.05 while leaving preparation unchanged. The attenuation becomes 0.81, t becomes 0.7938, and the raw expectation becomes 0.84535. More shots will clarify that lower raw performance; they cannot restore the original expectation. The appropriate intervention concerns measurement quality or an explicitly validated correction method.
This example separates physical effects from estimation precision. It also explains why a physical error budget and a service availability error budget need different definitions. One concerns the workload's scientific behavior; the other concerns eligible service events and their operational success criteria.
Avoid adding incompatible error numbers
If a toy process has m independent opportunities for a specified Bernoulli failure, each with probability e, the probability of at least one failure is 1 − (1 − e)^m. The approximation me is useful when me is small. Neither formula licenses treating every reported gate infidelity as an independent event that destroys the algorithm.
Coherent errors can accumulate with circuit structure. Correlated faults violate independence assumptions. Some faults cancel, some change an observable weakly, and some are detected or corrected. A workload budget should therefore use a model or measured sensitivity appropriate to its operations and acceptance rule.
Exercise: what can ten times more shots fix?
A team observes a persistently low raw score after readout calibration changes. It proposes ten times more shots, claiming this will improve the state fidelity by roughly a factor of √10. Evaluate the claim.
Answer. Under independent stationary sampling, statistical widths often decrease proportionally to the inverse square root of shot count. State fidelity does not improve through that arithmetic. If the change is in readout, the team must characterize the new measurement process and keep its context separate. More shots can estimate the changed score more precisely while leaving its bias intact.
Calibration and execution validity explains how to decide whether a measurement model still describes the conditions of a run.