Gaurav Gyawali, Henry Shackleton, Zhu-Xi Luo, Michael Lawler (Mar 20 2025).
Abstract: Finding good quantum codes for a particular hardware application and determining their error thresholds is central to quantum error correction. The threshold defines the noise level where a quantum code switches between a coding and a no-coding \emphphase. Provided sufficiently frequent error correction, the quantum capacity theorem guarantees the existence of an optimal code that provides a maximum communication rate. By viewing a system experiencing repeated error correction as a novel form of matter, this optimal code, in analogy to Jaynes's maximum entropy principle of quantum statistical mechanics, \emphdefines a phase. We explore coding phases from this perspective using the Open Random Unitary Model (ORUM), which is a quantum circuit with depolarizing and dephasing channels. Using numerical optimization, we find this model hosts three phases: a maximally mixed phase, a ``
Z2 code'' that breaks its U(1) gauge symmetry down to
Z2, and a no-coding phase with first-order transitions between them and a novel \emphzero capacity multi-critical point where all three phases meet. For the
Z2 code, we provide two practical error correction procedures that fall short of the optimal codes and qualitatively alter the phase diagram, splitting the multi-critical point into two second-order coding no-coding phase transitions. Carrying out our approach on current noisy devices could provide a systematic way to construct quantum codes for robust computation and communication.
