Phases of decodability in the surface code with unitary errors

Abstract: The maximum likelihood (ML) decoder in the two-dimensional surface code with generic unitary errors is governed by a statistical mechanics model with complex weights, which can be simulated via (1+1)D transfer matrix contraction. Information loss with an increasing error rate manifests as a ferromagnetic-to-paramagnetic transition in the contraction dynamics. In this work, we establish entanglement as a separate obstruction to decoding; it can undergo a transition from area- to volume-law scaling in the transfer matrix contraction with increasing unitary error rate. In particular, the volume-law entanglement can coexist with ferromagnetic order, giving rise to a phase in which the encoded information is retained yet is effectively undecodable. We numerically simulate the ML decoding in the surface code subject to both single- and two-qubit Pauli-X rotations and obtain a phase diagram that contains a ferromagnetic area-law, a paramagnetic volume-law, and a potential ferromagnetic volume-law phase. We further show that, starting from the paramagnetic volume-law phase, tilting the single-qubit rotation away from the X-axis couples the stat-mech models for X and Z errors and can lead to a ferromagnetic volume-law phase in which, although Z errors remain correctable, the encoded classical information is hard to recover. To perform numerical simulations, we develop an algorithm for syndrome sampling based on the isometric tensor network representation of the surface code.