Single-Shot and Few-Shot Decoding via Stabilizer Redundancy in Bivariate Bicycle Codes

TL;DR

This paper analyzes bivariate bicycle quantum LDPC codes, revealing how a polynomial determines their fault tolerance and stabilizer redundancy, and introduces algebraic design principles for improved single-shot decoding performance.

Contribution

It establishes a direct link between the polynomial g(z) and code properties, deriving bounds and providing new design rules for quantum codes with enhanced fault tolerance.

Findings

Polynomial g(z) governs stabilizer redundancy and syndrome code structure.

Constructed small codes with improved syndrome distance and evaluated decoding.

Identified a trade-off between quantum rate and syndrome distance affecting single-shot performance.

Abstract

Bivariate bicycle (BB) codes are a prominent class of quantum LDPC codes constructed from group algebras. While the logical dimension and quantum distance of \emph{coprime} BB codes are known to be determined by a greatest common divisor polynomial $g(z)$, the properties governing their fault tolerance under noisy measurement have remained implicit. In this work, we prove that this same polynomial $g(z)$ dictates the code's stabilizer redundancy and the structure of the classical \emph{syndrome codes} required for single-shot decoding. We derive a strict equality between the quantum rate and the stabilizer redundancy density, and we provide BCH-like bounds on the achievable single-shot measurement error tolerance. Guided by this framework, we construct small coprime BB codes with significantly improved syndrome distance ($d_S$) and evaluate them using BP+OSD. Our analysis reveals a structural bottleneck: within the coprime BB ansatz, high quantum rate imposes an upper bound on syndrome distance, limiting single-shot performance. These results provide concrete algebraic design rules for next-generation 2BGA codes in measurement-limited architectures.