Error-Correcting Phases of the Surface Code Under Coherent Errors

The surface code, with its high encoding threshold, is a promising candidate for quantum memory. Yet, its threshold under coherent errors still lacks a complete understanding. In this work, we study the maximum-likelihood decoding in the surface code under single-qubit unitary rotations that create one type of anyon excitation. This decoding problem is known to be governed by the Chalker-Coddington network model in Class D. We show that, surprisingly, the decoding phase diagram for the optimal decoder which knows the rotation angle of the error operator differs from that of the suboptimal decoder which has an imperfect estimation. Specifically, we derive an effective description as a non-linear sigma model with target space SO(2n)/U(n) and show that the optimal and suboptimal decoder are associated with distinct replica limits n → 1, 0. This predicts a decoding transition for the suboptimal decoder corresponding the metal-insulator transition in the network model described by the non-linear sigma model. However, the metal fixed point becomes unstable for the optimal decoder, hinting at a possible decodable phase existing up to the maximum rotation angle. We verify these predictions with extensive numerical simulations using a maximum-likelihood decoding algorithm we develop.