Quantum Bicycle LDPC Codes with High kd2/n from Divisor-Driven Search

Research Approach

This paper reformulates the construction of quantum bicycle (two-block circulant) low-density parity-check (LDPC) codes by mapping them into the polynomial ring F2[x]/(x^l - 1). By shifting from group-algebra formulations to this polynomial domain, the authors demonstrate that self-orthogonality is guaranteed automatically. Furthermore, the quantum dimension can be determined precisely by calculating the greatest common divisor (gcd) of the defining polynomials, and the minimum distance can be verified using the Calderbank correspondence to additive codes over F4. This algebraic structure allows the authors to implement a pre-filtered search algorithm that prunes the search space significantly before performing expensive matrix computations.

The authors successfully applied this divisor-driven search to identify several high-performing quantum codes. Notably, they discovered a [[66, 20, 7]] code that achieves a figure of merit (kd^2/n) of 14.85, surpassing the [[144, 12, 12]] bivariate bicycle code (kd^2/n = 12) while using less than half the block length. The search also recovered known short codes and produced a family of new codes at n=90, such as [[90, 16, 6]], [[90, 18, 6]], and [[90, 20, 6]]. Additionally, an exhaustive census at n=48 helped delineate the boundaries of the framework, proving that certain distance properties are unattainable within specific symmetric coset families, thereby identifying the limits of the cyclic construction.

Quantum LDPC codes are essential for reducing the physical qubit overhead required for fault-tolerant quantum computing. By replacing complex group-theoretic search methods with a systematic, polynomial-based approach, this work provides a more efficient route to discovering high-quality quantum codes. The ability to control dimension and certify distance algebraically before building large matrices makes it possible to explore parameter regimes that were previously computationally inaccessible, offering a practical path toward more efficient quantum error correction architectures.