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. One of the key issues is understanding how topological quantum memory is affected by realistic coherent noises, such as random rotation noise and amplitude damping noise. In this work, we investigate the intrinsic error threshold of the two-dimensional toric code, a paradigmatic topological quantum memory, under these coherent noises by employing both analytical and numerical methods based on the doubled Hilbert space formalism. A connection between the mixed-state phase of the decohered toric code and a non-Hermitian Ashkin-Teller-type statistical mechanics model is established, and the mixed-state phase diagrams under the coherent noises are obtained. We find remarkable stability of mixed-state topological order under random rotation noise with axes near the Y-axis of qubits. We also identify intriguing extended critical regions at the phase boundaries, highlighting a connection with non-Hermitian physics. The upper bounds for the intrinsic error threshold are determined by these phase boundaries, beyond which quantum error correction becomes impossible.