Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit. In this setting, we show that Maximum Probability Error (MPE) decoding and Maximum Likelihood (ML) decoding for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for MPE decoding, and from #SAT for ML decoding, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for MPE and ML decoding. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for MPE and ML decoding of arbitrary stabilizer codes with independent $X$ and $Z$ noise.

翻译：实际量子计算机将受到复杂、依赖于量子比特的噪声影响，而非简单的噪声（如所有量子比特具有相同强度的去极化噪声）。若解码算法能利用关于特定噪声的先验信息，我们能更有效地进行量子纠错。这促使我们考虑表面码解码的复杂度问题，其中解码问题的输入不仅包括综合征测量结果，还包括以每个量子比特的单量子比特泡利误差概率形式给出的噪声模型。在此设定下，我们证明表面码的最大概率误差（MPE）解码和最大似然（ML）解码分别是NP-hard和#P-hard。通过展示如何将布尔公式转化为依赖于量子比特的泡利噪声模型和一组编码公式可满足性属性的综合征，我们从SAT（针对MPE解码）和#SAT（针对ML解码）直接进行了归约。我们还给出了MPE和ML解码的近似难度结果。这些是平均情况下与许多高效表面码解码器正确的经验事实不矛盾的最坏情况难度结果（即，对于大多数综合征集合和大多数合理噪声模型）。这些难度结果与已知的具有独立$X$和$Z$噪声的任意稳定子码的MPE和ML解码的难度结果类似。