Mixed-State Topological Order under Coherent Noises

Mixed-state phases of matter under local decoherence have recently garnered significant attention due to the ubiquitous presence of noise in current quantum processors. Given the coherent nature of realistic noise, we investigate the two-dimensional toric code, a paradigmatic topological quantum memory, subject to two types of local coherent noise: random rotation noise and amplitude damping noise. By employing both analytical and numerical calculations based on the doubled Hilbert space formalism, we establish a connection between the mixed-state phase of the decohered toric code under these coherent noises and the Ashkin-Teller-type statistical mechanics model. We chart the phase diagrams for the two local coherent noises. The mixed-state phase boundaries identified here provide upper bounds for the intrinsic error threshold of the toric code under these coherent noises, beyond which quantum error correction becomes impossible.