Average-Case and Smoothed Near-Optimality for Color-Code Decoding

Abstract

Minimum-weight decoding for two-dimensional color codes is NP-hard (Walters and Turner 2026), motivating the search for approximation guarantees beyond worst-case exact decoding. We study a block-based decoder for triangular color-code lattices. The decoder satisfies the deterministic additive guarantee \(\lvert E_{\mathrm{alg}}\rvert \leq \operatorname{OPT}(S)+O(n/τ)\), where \(n\) is the number of vertices and \(τ\) is the wall spacing. We show that this additive guarantee becomes a near-optimal multiplicative guarantee under natural noise models. For constant-rate i.i.d. face noise and constant local degree, choosing \(τ=Θ(ε^{-1})\) gives a \((1+ε)\)-approximation with probability \(1-\exp(-Ω(n))\), in time \(n2^{O(ε^{-1})}\). We also prove a smoothed analogue: the same near-optimality guarantee holds when an arbitrary adversarial error pattern is perturbed by independent constant-rate noise. Finally, in the low-probability regime \(p=o(1/\log^2 n)\), the syndrome decomposes into small active regions with high probability, allowing independent component-wise decoding and yielding an exact minimum-weight correction in time \(n2^{O((\log n)^{3/2})}\). These results show that, despite worst-case hardness, color-code decoding admits strong average-case, smoothed, and sparse-regime guarantees.