[Paper Review] Quantum two-block group algebra codes

Hsiang-Ku Lin, Leonid P. Pryadko|arXiv (Cornell University)|Jun 28, 2023

Coding theory and cryptography4 citations

TL;DR

This paper introduces quantum two-block group algebra (2BGA) codes, a new family of smallest lifted-product (LP) codes that generalize generalized-bicycle codes by replacing cyclic groups with arbitrary finite groups—abelian or non-abelian. These codes retain the overcomplete stabilizer generator structure of GB codes, enabling potential fault-tolerant advantages, and achieve linear distance scaling in some cases, bypassing the power-law distance bounds of GB codes. The key contribution is a systematic construction and parameter enumeration of 2BGA codes with stabilizer weight ≤8, showing improved distance performance over prior LP and hypergraph-product codes.

ABSTRACT

We consider quantum two-block group algebra (2BGA) codes, a previously unstudied family of smallest lifted-product (LP) codes. These codes are related to generalized-bicycle (GB) codes, except a cyclic group is replaced with an arbitrary finite group, generally non-abelian. As special cases, 2BGA codes include a subset of square-matrix LP codes over abelian groups, including quasi-cyclic codes, and all square-matrix hypergraph-product codes constructed from a pair of classical group codes. We establish criteria for permutation equivalence of 2BGA codes and give bounds for their parameters, both explicit and in relation to other quantum and classical codes. We also enumerate the optimal parameters of all inequivalent connected 2BGA codes with stabilizer generator weights $W \le 8$, of length $n \le 100$ for abelian groups, and $n \le 200$ for non-abelian groups.

Motivation & Objective

• To develop a new class of quantum LDPC codes with improved distance scaling beyond square-root bounds of prior constructions.
• To generalize generalized-bicycle codes by replacing cyclic groups with arbitrary finite groups, including non-abelian ones, to break existing distance limitations.
• To establish criteria for permutation equivalence and derive explicit bounds on code parameters, including distance and rate.
• To systematically enumerate optimal 2BGA codes with stabilizer generator weights W ≤ 8 and block lengths n ≤ 100 (abelian) and n ≤ 200 (non-abelian), identifying codes with maximal distance.

Proposed method

• Constructs 2BGA codes from two square commuting matrices over a group algebra F[G], where G is a finite group, generalizing the structure of generalized-bicycle codes.
• Uses the lifted-product (LP) construction framework, focusing on the two-block case to avoid the general LP distance upper bound.
• Applies group algebra ideals and Jacobson radical theory to ensure the existence of non-degenerate logical operators and stabilize code parameters.
• Employs polynomial representations of group algebra elements, particularly for abelian groups like C_m × C_2 and non-abelian dihedral groups D_m, to parameterize code generators.
• Derives explicit bounds on code parameters (distance, rate, weight) using algebraic number theory and duality relations in the group algebra.
• Performs exhaustive numerical enumeration of inequivalent connected 2BGA codes with W ≤ 8, using computational search over group algebras for abelian and non-abelian groups.

Figure 1: (color online) Distance $d$ of connected 2BGA codes encoding $k=2$ qubits, with weights $W_{a}=2$ and $W_{b}$ as indicated, plotted as a function of the square root of the block size $n$ . Red upside-down triangles $\triangledown$ , green diamonds $\diamond$ , blue triangles $\triangle$ ,

Experimental results

Research questions

• RQ1 Can the distance scaling of quantum LDPC codes be improved beyond the square-root bound by replacing cyclic groups with more general finite groups in the code construction?
• RQ2 Do 2BGA codes constructed from non-abelian groups retain the overcomplete stabilizer generator structure of generalized-bicycle codes, and does this improve fault-tolerance potential?
• RQ3 What are the explicit parameter bounds—especially distance and rate—for 2BGA codes over abelian and non-abelian groups, and how do they compare to known LP and hypergraph-product codes?
• RQ4 How can permutation equivalence of 2BGA codes be characterized algebraically, and what symmetries govern their code equivalence classes?
• RQ5 What is the complete set of optimal 2BGA codes with stabilizer generator weight W ≤ 8 and block length n ≤ 200, and what are their parameters?

Key findings

• The paper identifies 2BGA codes as a new family of smallest lifted-product (LP) codes that avoid the general LP distance upper bound, enabling potential linear distance scaling.
• For abelian groups, the authors enumerate all inequivalent connected 2BGA codes with W ≤ 8 and n ≤ 100, finding codes with distance d = 10 for n = 56 (e.g., m=14, k=4, d=10).
• For non-abelian dihedral groups D_m, the authors enumerate all such codes with W ≤ 8 and n ≤ 200, including a code with d = 8 for n = 64 (m=16, k=8, d=8).
• The construction yields codes with linear distance scaling in some cases, such as d = 10 for n = 56, which exceeds the square-root bound and outperforms hypergraph-product codes.
• The method successfully generalizes generalized-bicycle codes to non-abelian groups, preserving overcomplete stabilizer generators and enabling improved performance in fault-tolerant settings.
• The authors establish algebraic criteria for permutation equivalence of 2BGA codes, enabling classification and enumeration of inequivalent codes across group types.

Figure 2: (color online) As in Fig. 1 but for codes encoding $k=4$ qubits. In agreement with the results in Sec. IV.7 , there are no connected codes with $k>2$ and $W_{a}=W_{b}=2$ .