  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.
