  A fault-tolerant quantum computer will be supported by a classical decoding
system interfacing with quantum hardware to perform quantum error correction.
It is important that the decoder can keep pace with the quantum clock speed,
within the limitations on communication that are imposed by the physical
architecture. To this end we propose a local `pre-decoder', which makes greedy
corrections to reduce the amount of syndrome data sent to a standard matching
decoder. We study these classical overheads for the surface code under a
phenomenological phase-flip noise model with imperfect measurements. We find
substantial improvements in the runtime of the global decoder and the
communication bandwidth by using the pre-decoder. For instance, to achieve a
logical failure probability of $f = 10^{-15}$ using qubits with physical error
rate $p = 10^{-3}$ and a distance $d=22$ code, we find that the bandwidth cost
is reduced by a factor of $1000$, and the time taken by a matching decoder is
sped up by a factor of $200$. To achieve this target failure probability, the
pre-decoding approach requires a $50\%$ increase in the qubit count compared
with the optimal decoder.
