Color codes are an important family of topological quantum error-correcting codes, in part because they support a richer set of transversal logical gates than surface codes. Their practical use, however, requires efficient decoding. Minimum-weight decoding is natural under independent noise, but Walters and Turner recently proved that it is NP-hard for the two-dimensional color code. Subsequent independent work of Walters and of Gu, Wang, and Kubica gave polynomial-time approximation schemes; in terms of lattice size \(N\), their running times are \(O(N^{2+528/ε})\) and \(N(\log N)^{O(1/ε)}\), respectively, for a \((1+ε)\)-approximation. We ask whether stronger guarantees are possible for random instances generated by a natural iid noise model. For triangular color-code lattices with \(N\) faces and iid face-error probability \(p\), we obtain near-linear-time algorithms throughout the noise range considered. If \(p=o(1/\log^2 N)\), we compute an exact minimum-weight correction in deterministic \(N^{1+o(1)}\) time with high probability. If \(p=Ω(1/\log^2 N)\), with \(p\) bounded away from \(1/2\), we compute a \(1+o(1)\)-approximation in deterministic \(N^{1+o(1)}\) time with high probability. When the syndrome is supplied sparsely and has size \(s\), the corresponding high-probability running times improve to \((1+s)N^{o(1)}\) and \(s^{1+o(1)}\), respectively. The algorithms exploit different random structure at different noise scales: localization into small exact subproblems in the sparse regime, peeling of locally unambiguous errors at intermediate densities, and a block decomposition whose additive loss is negligible relative to the typical optimum at larger densities. Together, these results give a near-linear average-case counterpart to recent worst-case hardness and approximation results for color-code decoding.

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