We study exact decoding for the toric code and for planar and rotated surface codes under the standard independent \(X/Z\) noise model, focusing on Separate Minimum Weight (SMW) decoding and Separate Most Likely Coset (SMLC) decoding. For the SMW decoding problem, we show that an \(O(n^{3/2}\log n)\)-time decoder is achievable for surface and toric codes, improving over the \(O(n^{3}\log n)\) worst-case time of the standard approach based on complete decoding graphs. Our approach is based on a local reduction of SMW decoding to the minimum weight perfect matching problem using Fisher gadgets, which preserves planarity for planar and rotated surface codes and genus~\(1\) for the toric code. This reduction enables the use of Lipton--Tarjan planar separator methods and implies that SMW decoding lies in \(\mathrm{NC}\). For SMLC decoding, we show that the planar surface code admits an exact decoder with \(O(n^{3/2})\) algebraic complexity and that the problem lies in \(\mathrm{NC}\), improving over the \(O(n^{2})\) algebraic complexity of Bravyi \emph{et al.} Our approach proceeds via a dual-cycle formulation of coset probabilities and an explicit reduction to planar Pfaffian evaluation using Fisher--Kasteleyn--Temperley constructions. The same complexity measures apply to SMLC decoding of the rotated surface code. For the toric code, we obtain an exact polynomial-time SMLC decoder with \(O(n^{3})\) algebraic complexity. In addition, while the SMLC formulation is motivated by connections to statistical mechanics, we provide a purely algebraic derivation of the underlying duality based on MacWilliams duality and Fourier analysis. Finally, we discuss extensions of the framework to the depolarizing noise model and identify resulting open problems.

翻译：我们研究了标准独立 \(X/Z\) 噪声模型下环面码、平面表面码及旋转表面码的精确解码问题，重点关注分离最小权重解码与分离最似然陪集解码。针对分离最小权重解码问题，我们证明了表面码与环面码可实现 \(O(n^{3/2}\log n)\) 时间复杂度的解码器，较之基于完全解码图的标准方法 \(O(n^{3}\log n)\) 最坏情况时间复杂度有所提升。我们的方法基于利用 Fisher 构件将分离最小权重解码局部归约为最小权重完美匹配问题，该归约对平面与旋转表面码保持平面性，对环面码保持亏格~\(1\)。此归约使得 Lipton--Tarjan 平面分隔器方法得以应用，并表明分离最小权重解码属于 \(\mathrm{NC}\) 类。对于分离最似然陪集解码，我们证明平面表面码存在具有 \(O(n^{3/2})\) 代数复杂度的精确解码器，且该问题属于 \(\mathrm{NC}\) 类，改进了 Bravyi 等人 \(O(n^{2})\) 的代数复杂度结果。我们的方法通过陪集概率的对偶环表述，并利用 Fisher--Kasteleyn--Temperley 构造显式归约为平面 Pfaffian 计算。相同的复杂度度量适用于旋转表面码的分离最似然陪集解码。对于环面码，我们获得了具有 \(O(n^{3})\) 代数复杂度的精确多项式时间分离最似然陪集解码器。此外，尽管分离最似然陪集解码的提出受到统计力学关联的启发，我们基于 MacWilliams 对偶性与傅里叶分析，给出了底层对偶关系的纯代数推导。最后，我们讨论了该框架向去极化噪声模型的扩展，并指出了由此产生的开放性问题。