The source of hardware-tailored codes and coding phases

A central challenge in quantum error correction is identifying powerful quantum codes tailored to specific hardware. This problem is hard because we cannot determine the noise models for our devices. Inspired by the quantum capacity theorem, we seek an optimal quantum source of information, namely the density matrix that degrades minimally when passed through a noisy channel. We explore this idea with the Open Random Unitary Model (ORUM). We find that the ORUM hosts three discrete regimes, the "maximally mixed source" phase, a "Z2 source" phase, and a no-coding phase. These phases exhibit first-order transitions among themselves and converge at a novel zero-capacity multicritical point. These results show a remarkable similarity between the quantum capacity theorem and Jaynes' maximum entropy principle of statistical mechanics. Using the Z2 source, we build two codes, a classical cat code capable of correcting all the dephasing errors and a concatenated cat code capable of correcting all errors up to a distance d=min(m,N) and reduces to Shor's 9-qubit code for m=N=3. Applying our approach to current noisy devices could provide a systematic method for constructing quantum codes for robust computation and communication.