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.

翻译：二维色码的最小权重解码是NP难的（Walters and Turner 2026），这促使人们寻求超越最坏情况精确解码的近似保证。我们研究了一种针对三角形色码格子的分块解码器。该解码器满足确定性加性保证 \(\lvert E_{\mathrm{alg}}\rvert \leq \operatorname{OPT}(S)+O(n/τ)\)，其中 \(n\) 是顶点数，\(τ\) 是墙间距。我们证明，在自然噪声模型下，这一加性保证可转化为近最优的乘性保证。对于恒定速率独立同分布的面噪声和恒定局部度数，选择 \(τ=Θ(ε^{-1})\) 可在时间 \(n2^{O(ε^{-1})}\) 内以概率 \(1-\exp(-Ω(n))\) 给出 \((1+ε)\)-近似。我们还证明了其光滑版本：当任意对抗性错误模式被独立恒定速率噪声扰动时，相同的近最优性保证依然成立。最后，在低概率区间 \(p=o(1/\log^2 n\) 内，综合征以高概率分解为小的活跃区域，从而允许独立的分量级解码，并在时间 \(n2^{O((\log n)^{3/2})}\) 内得到精确的最小权重修正。这些结果表明，尽管最坏情况下存在难度，色码解码在平均情形、光滑情形和稀疏情形下仍具备强保证。