Quantum two-block non-abelian group codes

We study numerically and analytically a class of quantum two-block CSS codes with generator matrices H_X = (A,B), H_Z = (B^T,A^T), where commuting square matrices A, B of size m × m are associated with elements of the group algebra F₂G, and G is a group of order m. These codes are the shortest lifted-product codes [1] that can be obtained from the group G. Also, they contain all generalized-bicycle (GB) codes obtained when G is the (abelian) group of cyclic permutations. The two-block group code (TBGC) ansatz is of interest as a method to construct short quantum LDPC codes without the upper distance bound of more regular GB codes [2]. Analytically, we show that non-trivial TBGCs with stabilizer generators of weight w = 4 can be always expressed as direct sums of GB codes. We also compute optimal parameters of TBGCs associated with non-abelian groups of orders m ≤ 50 by exhaustive enumeration of all such codes.

[1] P. Panteleev and G. Kalachev, “Asymptotically good quantum and locally testable classical LDPC codes,” arXiv:2111.03654

[2] R. Wang and L. P. Pryadko “Distance bounds for generalized bicycle codes,” arXiv:2203.17216