Univariate Bicycle Quantum LDPC Codes: Explicit Logical Structure and Distance Bounds

2026-05-13

AI Summary

Quantum LDPC code design faces challenges including ambiguous logical structure, difficulty in analyzing minimum distance, and construction complexity. This work proposes univariate bicycle (UB) codes—a structured subclass of generalized bicycle codes—that leverage Frobenius relations to reduce the design space from bivariate to univariate polynomial search. For the first time, the algebraic structure of their logical co-set space is explicitly characterized. By establishing a connection between logical representatives and cycle densities of circulant matrices, an upper bound on the minimum distance is derived. In the short-to-moderate blocklength regime—ranging from several hundred to approximately $10^3$—UB codes either outperform or match the performance of existing generalized and bivariate bicycle codes, demonstrating strong competitiveness while preserving stringent algebraic constraints.

Abstract

We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a two-polynomial search in GB codes to a single-polynomial search, while preserving sparsity. We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices. Finally, simulation results for short- to medium-length UB codes (block lengths ranging from a few hundred to approximately $10^3$) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.

Problem

• quantum LDPC codes
• generalized bicycle codes
• minimum distance
• logical operators
• code construction

Innovation

• univariate bicycle codes
• quantum LDPC codes
• logical operator parametrization
• minimum distance bounds
• circulant matrices