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

TL;DR

This paper introduces univariate bicycle quantum LDPC codes with explicit logical structures, providing algebraic characterization and distance bounds, and demonstrates their competitive performance through simulations.

Contribution

It presents a new subclass of quantum LDPC codes with simplified design, explicit logical structure, and proven distance bounds, advancing quantum error correction methods.

Findings

Explicit algebraic characterization of logical operators.

Derived upper bounds on minimum distance.

Simulation results show competitive performance.

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.