Fig. 2. Optimal code-capacity thresholds over all single-qubit Pauli channels.

Threshold estimates p_c are found using approximate maximum-likelihood decoding for a the XZZX surface code and b the CSS surface code with open boundaries (as in Fig. 1b). The grey triangle represents a parametrisation of all single-qubit Pauli channels, where the centre corresponds to depolarising noise, the labeled vertices correspond to pure X and Z noise, and the third vertex corresponds to pure Y noise. For the XZZX code, estimates closely match the zero-rate hashing bound (not shown) for all single-qubit Pauli channels. For the CSS code, estimates closely match the hashing bound for Y-biased noise but fall well below for X- and Z-biased noise. All estimates use d × d codes with distances d ∈ {13, 17, 21, 25}.