Error suppression factor (Λ)

The error suppression factor Λ says whether a quantum error-correcting code improves as it grows: the factor by which logical error per QEC cycle drops when the code distance increases by two. Coined in the Martinis group’s repetition-code experiment (Kelly et al., 2014; Nature 519, 66 (2015)) and called “the key metrological figure of merit” by Martinis in 2015, it organizes Google’s QEC roadmap. Λ is a fitted metric rather than a standardized protocol (code family, distances, cycle counts, and decoder vary by paper), but the memory-scaling experiments producing it act as de facto benchmarks of below-threshold operation.

How it works

Run logical memory experiments at successive code distances (d = 3, 5, 7, say) on one processor, extract the logical error per round ε_d at each, and fit ε_d = C / Λ^((d+1)/2). Then Λ = ε_d / ε_{d+2} ≈ p_thr / p: Λ > 1 certifies below-threshold operation, and larger Λ means faster exponential suppression as the code scales.

Strengths and limitations

Λ compresses “are we below threshold, and by how much?” into one number, but it is decoder-dependent: the same Willow dataset yields Λ = 2.14 with a neural-network decoder, 2.04 with ensembled matching, and 2.0 ± 0.1 in real time (arXiv:2408.13687). The single-exponential fit assumes uniform, uncorrelated noise (leakage, correlated errors, and cosmic rays violate it), and a 2025 IBM heavy-hex surface-code study argues single-parameter Λ fits can mischaracterize code performance (arXiv:2510.18847). Definitions have drifted: repetition-code values (Λ_X = 3.18, Λ_Z = 2.99 in 2021) are not comparable to surface-code Λ, and the coinage is often misattributed to the 2021–2024 Google papers. Coherence gain instead scores a logical qubit against the best physical qubit; the stability experiment probes complementary time-like errors.

Notable results

Kelly et al. measured Λ = 3.2 in a nine-qubit repetition code. Google’s first surface-code scaling test saw d = 5 barely edge d = 3 (2.914% vs 3.028% per cycle; arXiv:2207.06431) before Willow’s below-threshold milestone, Λ = 2.14 ± 0.02 at d = 3/5/7 (Nature 638, 920 (2025)), and the first color-code value, Λ_3/5 = 1.56(4) (arXiv:2412.14256).