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 quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, 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 QMLD and DQMLD. 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 QMLD and DQMLD for arbitrary stabilizer codes with independent $X$ and $Z$ noise.

翻译：实际量子计算机将面临复杂且依赖于量子比特的噪声，而非所有量子比特强度相同的退极化噪声等简单噪声。若我们的解码算法能利用特定噪声存在的先验信息，则可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——此时解码问题的输入不仅包括症候测量结果，还包含以每个量子比特单比特Pauli误差概率形式呈现的噪声模型。在此设定下，我们证明表面码的量子最大似然解码（QMLD）与简并量子最大似然解码（DQMLD）分别属于NP-hard和#P-hard问题。我们通过展示如何将布尔公式转换为依赖于量子比特的Pauli噪声模型及一组编码该公式可满足性特性的症候，直接从SAT问题归约至QMLD，从#SAT问题归约至DQMLD。我们还给出了QMLD和DQMLD的近似难度结果。这些最坏情况下的难度结果与许多高效表面码解码器在平均情况下正确的经验事实并不矛盾（即对大多数症候集合和大多数合理噪声模型而言）。这些难度结果与已知的独立$X$和$Z$噪声下任意稳定子码的QMLD和DQMLD难度结果具有优美的类比性。