Distance bounds for generalized bicycle codes

TL;DR

This paper investigates the properties of generalized bicycle quantum codes, establishing bounds on their distance and analyzing their performance, which could enhance quantum error correction strategies.

Contribution

It introduces new upper and lower bounds on the distance of GB codes and provides an exhaustive enumeration revealing their distance scaling behavior.

Findings

Distance bounds for GB codes depend on code parameters.

Observed distance scales approximately as A(w) * n^{1/2} + B(w).

Enumeration confirms theoretical bounds and scaling behavior.

Abstract

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 ans\"atze, 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 $d\ge {\cal O}({n}^{1/2})$. We have also done 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 $A(w){n}^{1/2}+B(w)$, where $n$ is the code length and $A(w)$ is increasing with $w$.