Surface-code threshold ledger

Rule. A threshold number is meaningless without its elementary fault model, syndrome model, decoder, and normalization.

The regimes below are deliberately not merged into one “surface-code threshold.”

Code capacity, optimal

• Model and decoder: independent X or Z data errors, perfect syndrome, and class-probability/maximum-likelihood decoding.
• Reported value: about 10.9%.
• Provenance: Dennis–Kitaev–Landahl–Preskill.

Code capacity, minimum energy

• Model and decoder: the same perfect-syndrome setting, using the zero-temperature/minimum-chain approximation associated with matching.
• Reported value: 0.1031 ± 0.0001.
• Provenance: Wang–Harrington–Preskill.

Phenomenological spacetime

• Model and decoder: repeated data and syndrome errors in the random-plaquette gauge/matching model.
• Reported value: 0.0293 ± 0.0002 in the quoted symmetric convention.
• Provenance: Wang–Harrington–Preskill.

Scheduled circuit

• Model and decoder: Fowler's specified eight-step syndrome circuit and per-step stochastic fault model.
• Reported value: p_th=0.57% per step, roughly 4% per full eight-step cycle.
• Provenance: Fowler–Mariantoni–Martinis–Cleland.

Hardware fit

• Model and decoder: Google Willow rotated memories with the device's measured heterogeneous noise and real-time decoder.
• Reported value: Λ=2.14 ± 0.02 per d→d+2; distance seven at 0.143% ± 0.003% per cycle.
• Provenance: Google Quantum AI, 2024.

1. Why 10.9% and 10.3% differ

For a fixed syndrome, the probability of a logical class is a sum over all error chains in that class. In the random-bond Ising mapping this is a partition function. Optimal decoding compares free energies on the Nishimori line. A minimum-chain decoder compares energies and discards most of the entropy from nearly degenerate paths. The numerical gap is therefore a decoder/statistical-mechanics distinction within one perfect-syndrome model, not experimental uncertainty on one number.

2. Why 2.9% is not a gate threshold

Adding noisy syndrome bits creates a time direction and a three-dimensional matching/gauge problem. A phenomenological “data error per round” and “measurement-bit flip per round” do not specify physical preparation, CNOT, idle, leakage, reset, or readout locations. The resulting threshold is valuable for decoder physics but cannot be compared directly with a circuit-location threshold.

3. Why the 0.57% conversion uses eight steps

The cited Fowler simulation defines p per step and explicitly uses eight steps per cycle. Its quoted “roughly four percent per cycle” is the conversion for that model. A different six-layer circuit may also be a valid surface-code schedule, but one may not attach Fowler's eight-step conversion to it without rebuilding the noise accounting.

4. Scaling below threshold

For simple independent models, the leading low-p logical failure mechanism needs ⌈d/2⌉ faults, motivating an empirical form

p_L ≈ A (p/p_th)^⌈d/2⌉.

Increasing odd distance by two then contributes roughly one further factor of p/p_th, before prefactors, degeneracy, correlations, and finite-size effects. Hardware Λ is fitted from the complete device-and-decoder stack. It should not be identified exactly with p_th/p.

5. Citation discipline

When a new threshold is added to the Atlas, record:

1. data-error channel;
2. whether syndrome is perfect, phenomenological, or circuit-derived;
3. complete circuit-location convention if applicable;
4. decoder and whether it uses correlations/degeneracy;
5. normalization: per location, step, round, cycle, or unit time.

Without those five fields, retain the number in the literature sweep rather than promoting it into reader-facing content.