Phases of Floquet code under local decoherence

Floquet code is a dynamical quantum memory with a periodically evolving logical space. As a defining feature, the code exhibits an anyon automorphism after each period, giving rise to a nontrivial evolution of each logical state. In this paper, we study the Floquet code under local decoherence and perfect measurements and demonstrate that, below the decoherence threshold, the code is in a robust phase characterized by the anyon automorphism. We first derive the three-dimensional (3D) statistical mechanics model for the maximum likelihood decoder of the two-dimensional (2D) Floquet code under local Pauli decoherence. We identify a class of two-qubit Pauli channels under which the 3D statistical mechanics model becomes decoupled 2D models and obtain the threshold for such decoherence channels. We then propose a diagnostic of the anyon automorphism in the presence of local decoherence. We analytically show that this diagnostic distinguishes the Floquet code from the toric code under repeated syndrome measurements and undergoes a phase transition at the threshold.