Logical error rate per round

The logical error rate per round (written ε_L, or “error per cycle of error correction”) is the headline number of a quantum error-correction memory experiment: the probability per syndrome-extraction round that the decoded logical qubit suffers a logical flip. The per-round quantity appears in surface-code theory long before any hardware demonstration, but Google Quantum AI’s 2021 repetition-code experiment, reporting “reducing logical error per round by more than 100x”, made it the standard experimental reporting convention. A metric and convention rather than a benchmark protocol in its own right, it is the number that below-threshold claims and the error-suppression factor Λ are built from.

How it works

A memory experiment prepares a logical qubit in an eigenstate of a logical observable, runs n rounds of stabilizer measurement, then reads out the data qubits and decodes the full syndrome history to decide whether the logical observable flipped. Sweeping n over many shots gives a logical fidelity that decays exponentially with rounds; fitting the decay, F(n) ∝ (1 − 2ε)^n, yields the per-round logical error ε, typically averaged over X- and Z-basis memory. Comparing ε across code distances gives Λ.

Strengths and limitations

The metric is operational and decoder-inclusive: qubits, syndrome-extraction circuits, and the classical decoder must all perform for ε_L to fall. But conventions vary (“per round” vs “per cycle”, the fit model, X/Z averaging, and per-logical-qubit normalization in multi-logical-qubit codes), so cross-paper comparisons need care. It also scores idle memory only: the stability experiment probes the complementary spacelike task, and DecoderBench isolates the classical decoding side.

Notable results

Google’s below-threshold surface code reports 0.143% ± 0.003% error per cycle at distance 7 (Nature, 2025), following the distance-scaling groundwork of its 2022 predecessor. Microsoft Azure Quantum and Quantinuum report per-cycle logical error rates below physical baselines on trapped ions, extended to graph states of up to 12 logical qubits with a tesseract code; 2025 qLDPC and bosonic demonstrations report the same per-cycle figure.