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.

中文翻译：

---

使用 Pauli 噪声解码表面代码的硬度结果


真正的量子计算机将受到复杂的、依赖于量子比特的噪声的影响，而不是简单的噪声，例如所有量子比特具有相同强度的去极化噪声。如果我们的解码算法考虑到有关存在的特定噪声的先验信息，我们可以更有效地进行量子纠错。这促使我们考虑表面代码解码的复杂性，其中解码问题的输入不仅是综合症测量结果，而且还是每个量子比特的单量子比特 Pauli 错误概率形式的噪声模型。


在这种设置中，我们表明表面代码的量子最大似然解码 （QMLD） 和简并量子最大似然解码 （DQMLD） 分别是 NP 硬和 #P 硬。我们直接从 QMLD 的 SAT 和 DQMLD 的 #SAT 进行简化，展示了如何将布尔公式转换为依赖于量子比特的 Pauli 噪声模型和一组编码公式满足性特性的综合征。我们还给出了 QMLD 和 DQMLD 的近似硬度结果。这些是最坏情况的硬度结果，与许多有效的表面代码解码器在平均情况下（即，对于大多数综合症集和最合理的噪声模型）是正确的经验事实并不矛盾。这些硬度结果与具有独立 $X$ 和 $Z$ 噪声的任意稳定器代码的 QMLD 和 DQMLD 的已知硬度结果非常相似。