Optimal quantum subsystem codes in 2-dimensions

Understand

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 2-dimensions with parameters $[[N,K,D]]$ satisfying $N\le 2n_1n_2$, $K=k$, and $D=\min(d_1,d_2)$.

• These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi.
• 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 2-dimensional subsystem codes.
• One of these good constructions uses classical expander codes.