Distance Bounds for Generalized Bicycle Codes

Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in $D\le w-1$ dimensions, and lower existence bounds which give $d\ge \mathcal{O}\left(n^{1/2}\right)$. We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with $A\left(w\right)n^{1/2}+B\left(w\right)$, where n is the code length and $A\left(w\right)$ is increasing with w.