Phases of Floquet quantum memory under local decoherence

The Floquet code, introduced by Hastings and Haah, encodes quantum information in a periodically evolving logical space and features a non-trivial anyon automorphism after each period. In this work, we show that the signatures of the Floquet code are robust up to a finite threshold of local decoherence. We begin by deriving a 3d statistics mechanics model for the maximum likelihood decoder of the Floquet code and obtain a finite decoherence threshold for a specific class of two-qubit Pauli channels. Furthermore, we demonstrate that below this threshold, the Floquet code is in a mixed-state phase characterized by a quantum relative entropy that probes the anyon automorphism. We analytically show that the relative entropy undergoes a phase transition at the threshold and can distinguish the Floquet code from the encoding phase of the Toric code under repeated syndrome measurements.