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Logical randomized benchmarking

Error correction proposal

Randomized benchmarking lifted to the logical layer: random logical Clifford sequences with QEC cycles interleaved yield an average error per logical gate of an encoded qubit.

Measures
Average error per logical Clifford gate, from the decay of logical survival probability over random logical Clifford sequences (the original analysis also splits out correctable vs uncorrectable error probabilities)
Introduced by
Combes, Granade, Ferrie & Flammia (2017)
Also known as
LRB, Logical RB

Logical randomized benchmarking (LRB) transplants randomized benchmarking to the logical layer of an error-corrected device, proposed in 2017 by Combes, Granade, Ferrie & Flammia (arXiv:1702.03688). Like physical RB it isolates an average per-gate error (component-level in spirit), but the gates are logical Cliffords on a QEC-encoded qubit, so the score folds in the code’s syndrome-extraction and decoding machinery and it sits with the error-correction benchmarks here.

How it works

Sample random sequences of logical Clifford gates of increasing length, apply them to an encoded qubit with error-correction cycles interleaved, append the inverting Clifford, and measure whether the logical state survives. Fitting the decay of logical survival probability against sequence length yields the average error per logical Clifford gate; the original analysis additionally estimates the average probabilities of correctable and uncorrectable errors.

Strengths and limitations

LRB inherits RB’s virtues (insensitivity to state-preparation and measurement errors, and a single interpretable figure of merit) at the level users of a fault-tolerant machine actually care about. But its guarantees rest on assumptions that can fail: Ceasura, Iyer, Wallman & Pashayan showed that QEC machinery such as syndrome qubits coupled to a decoder can make the LRB signal oscillate rather than decay exponentially, breaking the standard exponential fit. The name is also used loosely for any RB run on encoded qubits, and experiments do not necessarily implement the full Combes et al. analysis. No dedicated open-source implementation exists.

Notable results

As a complete protocol, LRB remains a proposal: the 2017 paper is a v1-only preprint, never journal-published. Logical-gate RB in the broader sense has entered practice, though: Google’s color-code experiment used logical randomized benchmarking to report an average transversal logical Clifford error of 0.0027(3) (arXiv:2412.14256; Nature 645, 614 (2025)), and Quantinuum ran RB on a logical two-qubit gate in the [[7,1,3]] Steane code on H2-1 (arXiv:2404.08616). Compare logical error per round for logical memory rather than gates, and logical magic state benchmarking for the non-Clifford side.

Key papers

  • 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.
  • Logical error rate per round : The headline figure of QEC memory experiments: the probability per round of error correction that the decoded logical qubit suffers a logical flip.
  • Logical magic state benchmarking : Proposed protocols that estimate the infidelity of a prepared logical magic state to multiplicative precision using only fault-tolerantly implementable Clifford operations.