Readout
Assignment bias on the measurement step.
Mis-assigned bitstrings shift every expectation value asymmetrically. Modelled as a calibration matrix on the count vector. Readout twirling runs as a sub-primitive of the random-Clifford shadow path.
Tool 5 / Mitigation + Measurement Spine
Move the slider.
Move the slider.
The curve moves.
Zero-noise extrapolation, probabilistic error cancellation, readout calibration. Each technique returns a mitigated estimator with a bias bound and a sample-complexity certificate. The signed manifest names every primitive, every shot budget, every calibration cycle.
Figure 1. ZNE extrapolation, 4-qubit GHZ Z-parity, synthetic Markovian noise Ideal value E_true = 1.000
Section one
The four error classes.
Each class has a characteristic noise-curve shape. The spine treats each one with a different primitive and reports the assumption breaks in plain text.
Coherent
Miscalibrated rotation angle.
Over-rotation, control crosstalk, drift. Survives a single circuit, does not survive a twirl. Cycle benchmarking Pauli-twirls each native cycle and converts the residue into a sparse Pauli-Lindblad rate set with a reported fit residual.
Incoherent
Exponential decay under a Lindblad envelope.
Decoherence on the actual circuit. The cycle-benchmarking fit reports layer-resolved Pauli-Lindblad rates. The ZNE bias bound is computed against those rates, not against vendor-quoted depolarising numbers.
Leakage
Out of the qubit subspace. Spine refuses.
Population escapes to the second-excited state. The qubit assumption breaks. The analyticity diagnostic detects the departure, refuses the mitigated estimator, and reports the assumption break in plain text. A quiet correction would be worse than a refusal.
Leakage is a hard exclusion. The spine refuses to extrapolate when the analyticity diagnostic detects population outside the qubit subspace.
Section two
Mitigation budget allocation.
How a fixed mitigation budget splits across primitives depends on the workload. Three illustrative presets, drawn from the workload taxonomy the spine routes against.
Variational ansatz
Many local observables, depth-bounded ansatz.
- ZNE 25%
- Shadow ensemble 50%
- Readout twirling 15%
- Leakage detection 10%
Trotterised circuit
Deep dynamics, few global observables.
- ZNE 55%
- Shadow ensemble 15%
- Readout twirling 20%
- Leakage detection 10%
Random circuit
Cross-entropy with stated attack-resistance list.
- ZNE 10%
- Shadow ensemble 25%
- Readout twirling 20%
- Leakage detection 45%
Section three
The spine pipeline.
One central spine. Five outward feeds. Each sibling tool reads its named payload from the spine at its own measurement gate. The diagram draws what the spine claim says.
Section four
Bias-bound derivation.
For each mitigation primitive, the manifest names the bias-bound formula, the variance growth factor, and the sample-complexity certificate. One bound per method, one assumption regime per bound.
Richardson
|E_mit − E_true| ≤ A · λ_max^(N+1) · γ_max^(N+1) · D
N order of the Lindblad expansion truncated by the Richardson coefficients, γ_max maximum rate in the cycle-benchmarking fit, λ_max the largest noise factor in the schedule, D circuit depth, A a fit-residual coefficient.
Exponential
|E_mit − E_true| ≤ B · max_λ |R_2(λ)|
R_2 the second-order residual of the depolarising-fit assumption. Bound is tight when the cycle channel is depolarising. Loose otherwise; the analyticity diagnostic reports which regime the customer circuit is in.
Polynomial deg 2
|E_mit − E_true| ≤ C · max_k |coef_k| · λ_max^3
coef_k the polynomial-fit coefficients up to degree two, λ_max the largest noise factor. C bounded by the conditioning of the Vandermonde system on the noise-factor grid.
The shadow predictor carries a sample-complexity certificate of standard random-ensemble form. The number of snapshots required scales with the logarithm of the observable count, with the inverse square of the accuracy target, and with the maximum shadow-norm-squared over the observable set.
Section five
Vendor calibration cycle.
Every manifest records the vendor calibration timestamp at run time. Cadence varies by vendor. The spine records, the spine does not assume.
Vendor 01
IBM Heron r2
Median calibration cycle
Cadence Every 30 min
Vendor 02
IonQ Tempo
Operator-initiated calibration pass
Cadence Per shift
Vendor 03
Quantinuum H3
Continuous-recalibration regime
Cadence Continuous
Section six
What this does not claim.
The mitigation primitives shipped here are characterised on standard noise models with their bias bounds named. The spine does not claim a single technique dominates across all hardware. It does not extrapolate when leakage is detected. It does not replace error correction. It provides a documented, manifested, replayable mitigation envelope for the NISQ regime, with explicit assumption breaks per primitive.
Section seven
Questions.
- Question 01
How is the bias bound computed?
From the Lindblad expansion truncation order against the Pauli-Lindblad rates fit by the cycle-benchmarking step. The expansion order, the maximum fitted rate, and the maximum noise factor are reported alongside the value. If the analyticity diagnostic fails, the bound is reported as a lower bound only and the mitigated value is held back until the customer agrees to the conditional reading.
- Question 02
What if my observable set is too large for the shot budget?
The predictor reports the number of snapshots required by the standard random-ensemble bound and compares it to the budget. If the budget is insufficient, the predictor refuses to fit and recommends one of three actions: enlarge the budget, restrict the observable family, or move to the derandomised protocol for Pauli-string observables. The shot-budget check is a gate in the pipeline. Under-sampled runs do not ship.
- Question 03
Which backends does the spine support?
The spine ships against the customer circuit on a supported backend in the IBM Heron, Quantinuum H3, and IonQ Tempo families. Pulse-stretched ZNE runs where pulse-level access is available. Gate-folded ZNE runs as the digital fallback. The chosen branch is recorded in the manifest.
- Question 04
What does the spine NOT claim?
It does not replace fault-tolerant quantum error correction. It does not mitigate logical errors. It does not claim formal advantage over classical simulation. It claims one thing only: at a stated noise level, on a stated circuit class, the mitigated estimator has a stated bias bound, the shadow predictor has a stated sample-complexity certificate, and the noise profile has a stated bootstrap confidence interval. The assumption breaks are listed verbatim per primitive.