Optimal quantum subsystem codes in two dimensions
Theodore J. Yoder · Physical Review A · 2019
Given any two classical codes with parameters $[{n}_{1},k,{d}_{1}]$ and $[{n}_{2},k,{d}_{2}]$, we show how to construct a quantum subsystem code in two dimensions with parameters $[[N,K,D]]$ satisfying $N\ensuremath{\le}2{n}_{1}{n}_{2}, K=k$, and $D=min({d}_{1},{d}_{2})$. These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi [Phys. Rev. A 83, 012320 (2011)]. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound $KD=O(N)$ on the code parameters of two-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the additional advantage of a linear time quantum decoder based on the classical Sipser-Spielman flip decoder. Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge fixed to certain hypergraph product codes that do.