A matching threshold of 1.6×10−3 for the rotated surface code under circuit-level noise
In plain words
The rotated surface code, decoded by minimum-weight perfect matching, is a standard model of a fault-tolerant quantum memory: below a threshold error rate, its logical error rate falls exponentially as the code grows. Simulations put this threshold near 0.5–1% for circuit-level noise, while proofs had reached 1.7×10⁻⁴ (Dennis, Kitaev, Landahl and Preskill) and 7.4×10⁻⁴ (Fowler), each for its own circuit and noise model. This paper proves exponential decay for the standard syndrome-extraction circuit whenever every gate, idle step, preparation and measurement fails independently with probability at most 1.6×10⁻³. A local criterion for weighted walks gives 1.04×10⁻³ in closed form. The sharper value uses the minimality of the matching correction, with a potential function checked in exact arithmetic on all 2.5 million local configurations.
Abstract
We consider the rotated surface code of odd distance d with its standard syndrome-extraction circuit, under circuit-level depolarizing noise in which every gate, idle step, preparation and measurement fails independently with probability p, and decoded by minimum-weight perfect matching with the log-likelihood weights of single faults. We prove that a memory experiment with T rounds fails with probability at most 0.032 (T+1)(d+1) λd/(1−λ), where λ = 0.997 (p/p**)21/50, for every p ≤ p** = 1.6×10−3. Hence the logical error rate per round decays exponentially in d for all p ≤ p**. Earlier rigorous bounds for matching on surface-code circuits are 1.7×10−4, due to Dennis, Kitaev, Landahl and Preskill, and 7.4×10−4, due to Fowler. We keep the reduction of Dennis, Kitaev, Landahl and Preskill from a matching failure to a weighted self-avoiding path in the circuit-level decoding graph, and bound these paths with a local criterion for non-backtracking walks, which already gives 1.04×10−3 in closed form. The sharper value comes from the minimality of the correction along the path: we exclude the labellings of short stretches of the path into error and correction edges that contradict it, and count the remaining labelled walks with a potential function, which is checked in exact arithmetic on all local configurations. For a leaner schedule with fused measure-and-reset steps the same arguments give 1.53×10−3 and 2.4×10−3. Both arguments extend to error rates that depend on the type of operation. They certify exponential decay for error models built from the published component error rates of Quantinuum's Helios and H2 and of IBM's Heron r3 processor ibm_boston, mapped onto this circuit. Any argument that bounds the failure probability by the probability of an increasing event, ours included, is limited by the smallest such event that contains all failures. Monte Carlo estimates place its threshold for this circuit near 4×10−3. The computations that the proofs use are exact, and the fault table is cross-checked against Stim.