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The development of practical, high-performance decoding algorithms reducesthe resource cost of fault-tolerant quantum computing. Here we propose adecoder for the surface code that finds low-weight correction operators forerrors produced by the depolarising noise model. The decoder is obtained bymapping the syndrome of the surface code onto that of the color code, therebyallowing us to adopt more sophisticated color-code decoding algorithms.Analytical arguments and exhaustive testing show that the resulting decoder canfind a least-weight correction for all weight $d/2$ depolarising errors foreven code distance $d$. This improves the logical error rate by an exponentialfactor $O(2^{d/2})$ compared with decoders that treat bit-flip and dephasingerrors separately. We demonstrate this improvement with analytical argumentsand supporting numerical simulations at low error rates. Of independentinterest, we also demonstrate an exponential improvement in logical error ratefor our decoder used to correct independent and identically distributedbit-flip errors affecting the color code compared with more conventionalcolor-code decoding algorithms.

Minimising surface-code failures using a color-code decoder