The development of practical, high-performance decoding algorithms reduces
the resource cost of fault-tolerant quantum computing. Here we propose a
decoder for the surface code that finds low-weight correction operators for
errors produced by the depolarising noise model. The decoder is obtained by
mapping the syndrome of the surface code onto that of the color code, thereby
allowing us to adopt more sophisticated color-code decoding algorithms.
Analytical arguments and exhaustive testing show that the resulting decoder can
find a least-weight correction for all weight d/2 depolarising errors for
even code distance d. This improves the logical error rate by an exponential
factor O(2d/2) compared with decoders that treat bit-flip and dephasing
errors separately. We demonstrate this improvement with analytical arguments
and supporting numerical simulations at low error rates. Of independent
interest, we also demonstrate an exponential improvement in logical error rate
for our decoder used to correct independent and identically distributed
bit-flip errors affecting the color code compared with more conventional
color-code decoding algorithms