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Unitarity randomized benchmarking

Component-level

Randomized-benchmarking variant that fits the decay of output-state purity to measure how coherent a gate set's noise is, splitting error budgets into miscalibration and decoherence.

Measures
Unitarity, the coherence of gate-set noise, from the decay of output-state purity with random Clifford sequence length
Introduced by
Wallman, Granade, Harper & Flammia (2015)
Also known as
Unitarity RB, Purity Benchmarking, Purity RB, Purity Randomized Benchmarking, XRB, Extended Randomized Benchmarking

Unitarity randomized benchmarking asks not how large a gate set’s error is, but how coherent it is. Introduced by Wallman, Granade, Harper and Flammia in 2015, it reuses the random-Clifford machinery of standard randomized benchmarking but scores the coherence of the noise, which makes it as much a noise diagnostic as a performance benchmark. Combined with the ordinary RB error rate, it splits a gate’s error budget into coherent miscalibration and irreversible decoherence.

How it works

Random Clifford sequences of increasing length m are applied to one or a few qubits, and instead of a survival probability the experimenter estimates the output state’s purity (the squared length of its generalized Bloch vector) via state tomography or two-copy measurements. The average purity decays as A + B·u^(m−1), and the fitted base u is the unitarity: u = 1 for purely coherent (unitary) noise, smaller as the noise becomes stochastic. Wallman et al. show that u bounds the lowest infidelity reachable by recalibration alone: an RB error rate far above the u-implied floor flags calibration errors worth hunting, while a rate at the floor says decoherence dominates.

Strengths and limitations

Unitarity is a property of the noise, not a ranking: u = 1 means the noise is perfectly coherent, not that the gates are good. Purity estimation is the costly part (tomography or two copies), so published uses are mostly one and two qubits; Dirkse, Helsen and Wehner reduce the sample complexity for few-qubit Clifford gates. Two naming traps: “XRB” is the True-Q product name for this same protocol, not a distinct method, and speckle purity benchmarking measures the same coherent-versus-incoherent split from XEB data rather than from purity measurements after Clifford sequences.

Notable results

First demonstrated on a solid-state qubit in 2016, it is now routine for error budgeting: a 1110-hour purity-benchmarking noise study of a superconducting qubit appeared in APL Quantum in 2024, the protocol ships commercially as True-Q’s XRB, it is covered as a standard QCVV protocol in the Hashim et al. tutorial, and Rigetti’s forest-benchmarking carries an open-source implementation.

Key papers

Reference implementations

  • Randomized Benchmarking : Estimates average gate error from the exponential decay of survival probability over increasingly long random Clifford sequences, robust to state-preparation and measurement errors.
  • Speckle purity benchmarking : Reads the decoherence-limited fidelity of a gate straight out of cross-entropy benchmarking data by estimating state purity from the speckle contrast of measured bitstring probabilities.
  • Gate set tomography : Calibration-free tomography that reconstructs every gate, state preparation, and measurement in a gate set simultaneously and self-consistently, yielding predictive error models rather than a single score.