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

2026-06-09

AI Summary

Minimum-weight decoding for two-dimensional color codes is NP-hard in the worst case and lacks efficient approximation guarantees. This work proposes a deterministic, block-structured decoder that combines block decomposition, independent component decoding, and probabilistic analysis to achieve an additive approximation on triangular lattices. Under constant face noise and smooth adversarial perturbations, this approach yields a near-optimal multiplicative approximation. The study establishes the first strong approximation guarantees under average-case, smoothed, and sparse error models: at constant noise rates, it achieves a $(1+\varepsilon)$-approximation with exponentially high probability; at low error rates, it enables exact decoding in time $n\cdot2^{O((\log n)^{3/2})}$, thereby circumventing worst-case complexity barriers.

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.

Problem

• color-code decoding
• NP-hardness
• average-case analysis
• smoothed analysis
• minimum-weight decoding

Innovation

• color-code decoding
• average-case analysis
• smoothed analysis
• near-optimality
• minimum-weight decoding