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 and determining their error thresholds above which quantum information is unprotected. 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), a simplified model of a $N$-qubit quantum computer with competing depolarizing and dephasing channels as a stand-in for unitary gates and measurements. Through numerical optimization, we find that the ORUM hosts three discrete regimes, three"phases", the"maximally mixed source"phase, a"$\mathbb{Z}_2$ source"phase (where ORUM's $U(1)$ gauge symmetry is broken down to $\mathbb{Z}_2$), and a no-coding phase where all information is lost. 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 $\mathbb{Z}_2$ 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 $d=\text{min}(m,N)$ and reduces to Shor's 9-qubit code for $m=N=3$. Neither classical nor quantum code survives near the vicinity of the zero-capacity multicritical point in the source phase diagram. Applying our approach to current noisy devices could provide a systematic method for constructing quantum codes for robust computation and communication.