Fig. 8. Optimal threshold sample distribution and decoder convergence.

a Distribution of 211 samples over the surface of all single-qubit Pauli noise channels. To construct each threshold surface of Fig. 2, by code symmetry, thresholds are estimated for 111 of these samples. b Tensor-network decoder convergence for the 77 × 77 XZZX surface code with Z-biased noise, represented by shifted logical failure rate f_χ − f₂₄, as a function of truncated bond dimension χ at a physical error probability p near the zero-rate hashing bound for the given bias η. Each data point corresponds to 30,000 runs with identical errors generated across all χ for a given bias.