Hardness results for decoding the surface code with Pauli noise

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.