Existence and Characterization of Bivariate Bicycle Codes

Abstract

Encoding quantum information in a quantum error correction (QEC) code offers protection against decoherence and enhances the fidelity of qubits and gate operations. One of the fundamental challenges of QEC is to construct codes with asymptotically good parameters, i.e. a non-vanishing rate and relative minimum distance. Recently, bivariate bicycle (BB) codes have emerged as a promising candidate for such compact memory, though the exact tradeoff of the code parameters $[[n,k,d]]$ remained unknown. Despite recent proof that BB codes are asymptotically bad, they serve as candidate codes for near-term QEC beyond the surface code. In this article, we provide a more concrete characterisation of the performance of BB codes by identifying sufficient and necessary conditions for these codes to exist, using the fact that they are generated by finite field trinomials, and writing explicit formulas to calculate their code dimension by identifying common roots of these trinomials, using Gröbner bases.