Existence and Characterisation 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. Such codes provide compact quantum memory with low overhead and enhanced error correcting capabilities, compared to state-of-the-art topological error correction codes such as the surface or colour codes. 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. In this Article, we explore these codes by leveraging their ring structure, and predict their dimension as well as conditions on their existence. Finally, we highlight asymptotic badness. Though this excludes this subclass of codes from the search towards practical good low-density parity check (LDPC) codes, it does not affect the utility of the moderately long codes that are known, which can already be used to experimentally demonstrate better QEC beyond the surface code.