Speckle purity benchmarking

Speckle purity benchmarking (SPB) estimates how much of a gate’s error is decoherence, using no data beyond what cross-entropy benchmarking (XEB) already collects. It was defined in the two-qubit gate metrology section of the supplementary information to Google’s 2019 quantum-supremacy experiment, and, as an estimator applied to XEB measurement data rather than a standalone circuit protocol, it has become a standard ingredient of Google-style error budgeting.

How it works

Deep random circuits ideally produce bitstring probabilities following the Porter-Thomas distribution, whose wide spread makes a histogram of measured probabilities look “speckled.” Decoherence mixes the output toward the uniform distribution and washes the speckle out. SPB models the output as a depolarized pure state, ρ = p·|ψ⟩⟨ψ| + (1 − p)·I/D, whose purity is p², and extracts p from the speckle contrast: the variance of measured bitstring probabilities relative to the ideal Porter-Thomas variance. Fitting p against circuit depth turns an XEB dataset into a per-cycle purity decay: the fidelity the device would show if decoherence were its only error, with no experiments beyond XEB itself.

Strengths and limitations

The comparison it enables is the point: the gap between speckle purity and the measured XEB fidelity is attributed to coherent control error, the same coherent-versus-incoherent split as unitarity randomized benchmarking, but obtained from XEB data instead of Clifford-sequence purity measurements. Its limits follow from its assumptions: it is a statistical estimator, not a tomographic purity, valid when ideal outputs are Porter-Thomas and errors act like a depolarizing mixture, and it is by design insensitive to coherent errors. Many papers report the same quantity simply as “XEB purity” or “purity XEB,” without the word speckle.

Notable results

SPB is a standard part of Google’s gate metrology, used in the random-circuit-sampling phase-transition paper (Nature 634, 328 (2024)) and shipped in Cirq as purity_from_probabilities. It has also spread beyond Google, for example to CZ-gate error budgeting in tunable-coupler transmons.