Phases of decoding in the surface code with coherent errors

Two-dimensional surface code is a promising platform for quantum memory. Yet, its maximum-likelihood decoding threshold under coherent errors still lacks a complete understanding. In the first part of this talk, I will show that retaining quantum information does not necessarily imply efficient decodability using classical algorithms. This leads to an enriched decoding phase diagram. In the second part of the talk, I will revisit the simplest coherent error — single-qubit unitary rotations which create electric anyon excitations. In this case, I will develop an effective non-linear sigma model that governs the decoding. This effective theory leads to a surprising prediction of the phase diagram, which hinges on whether the decoder has a complete knowledge of the underlying error model. I will also present extensive numerical evidence to verify these predictions.