Breaking the bicycle frame: Coset-based quantum LDPC codes

2026-06-15

AI Summary

This work proposes a novel framework for constructing quantum low-density parity-check (LDPC) codes that overcomes the limitations of traditional two-block group algebra (2BGA) constructions. By replacing regular group actions with group actions on cosets of subgroups, the approach substantially expands the achievable parameter space. Integrating graph cover theory with algebraic coding techniques, the method yields new high-performance codes such as [[48,8,6]] and [[96,8,10]]. Accompanying these constructions are a maximum-filling check extraction schedule of depth w+2 and a BP-OSD decoding algorithm. Under a circuit-level noise model, the resulting codes achieve fault-tolerance thresholds of approximately 0.65% for weight-6 checks and 0.35% for weight-8 checks, rivaling the performance of state-of-the-art balanced product (BP) codes.

Abstract

Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$, $[[96,8,10]]$, and $[[224,12,16]]$, as well as weight-8 codes with parameters $[[84,16,8]]$, $[[112,16,10]]$, $[[128,16,12]]$, and $[[168,16,15]]$. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$, including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

Problem

• quantum LDPC codes
• coset-based construction
• two-block codes
• code parameters
• quantum error correction

Innovation

• coset-based construction
• quantum LDPC codes
• two-block group algebra
• syndrome extraction schedule
• graph covers