Surface-Code Threshold Estimates
The surface code tolerates physical error rates of roughly one percent under realistic circuit noise, the number that makes it a leading candidate.
What the threshold means
The threshold is the physical error rate below which increasing the code distance decreases the logical error rate. Above it, adding qubits makes things worse. The surface code's high threshold is the main reason it dominates near-term fault-tolerance plans: real hardware error rates are close to it, whereas most other codes demand far lower physical error rates.
The numbers
Under an idealized model where only data qubits suffer independent X and Z errors, the surface code's threshold is near ten percent. Under a realistic circuit-level model that includes faulty gates, faulty measurements, and repeated syndrome rounds, the threshold drops to roughly one percent, with the exact value depending on the noise model, decoder, and gate schedule.
- Code-capacity model (data errors only): threshold near 10 percent.
- Phenomenological model (data plus measurement errors): a few percent.
- Circuit-level model (everything faulty): about 1 percent.
- Better decoders and schedules push the circuit-level number modestly higher.
The gap between these numbers is a caution: quoting the code-capacity threshold overstates real performance by an order of magnitude. The circuit-level threshold is the one that matters, because it accounts for the fact that the syndrome-extraction circuit itself injects errors every round.
Below threshold, the logical error rate falls roughly as (p over p_threshold) raised to half the distance, so being a factor of a few below threshold yields fast improvement with each added row of qubits. This steep suppression, combined with a threshold near what hardware achieves, is exactly why the surface code sets the pace, even though its qubit overhead per logical qubit is large.