Abelian and non-abelian quantum two-block codes
Abstract
We discuss quantum two-block codes, a large class of CSS codes constructed from two commuting square matrices.Interesting families of such codes are generalized-bicycle (GB) codes and two-block group-algebra (2BGA) codes, where a cyclic group is replaced with an arbitrary finite group, generally non-abelian. We present code construction and give several expressions for code dimension, applicable depending on whether the constituent group is cyclic, abelian, or non-abelian. This gives a simple criterion for an essentially non-abelian 2BGA code guaranteed not to be permutation-equivalent to such a code based on an abelian group. We also give a lower bound on the distance which, in particular, applies to the case when a 2BGA code reduces to a hypergraph-product code constructed from a pair of classical group codes.
Cite
@article{arxiv.2305.06890,
title = {Abelian and non-abelian quantum two-block codes},
author = {Renyu Wang and Hsiang-Ku Lin and Leonid P. Pryadko},
journal= {arXiv preprint arXiv:2305.06890},
year = {2023}
}