Existence and Characterization of Bivariate Bicycle Codes
- 1Eindhoven University of Technology
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 withasymptotically goodparameters, 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.
Related
- Spectral Theory of Semisimple Bivariate Bicycle Codes
- Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes
- Metachecks in Bivariate Bicycle Codes: Syndrome Distance, Measurement Faults, and Repair Limits
- Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark
- Sequences of Bivariate Bicycle Codes from Covering Graphs
BibTeX
@article{postema2026existence,
title = {{Existence and Characterization of Bivariate Bicycle Codes}},
author = {J. J. Postema and S. J. J. M. F. Kokkelmans},
journal = {IEEE Transactions on Information Theory},
year = {2026},
month = jul,
doi = {10.1109/tit.2026.3713397},
url = {https://doi.org/10.1109/tit.2026.3713397},
note = {CC BY},
}
RIS
TY - JOUR
TI - Existence and Characterization of Bivariate Bicycle Codes
AU - J. J. Postema
AU - S. J. J. M. F. Kokkelmans
PY - 2026
DA - 2026/07/14
JO - IEEE Transactions on Information Theory
DO - 10.1109/tit.2026.3713397
UR - https://doi.org/10.1109/tit.2026.3713397
AB - 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 withasymptotically goodparameters, 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.
SN - 0018-9448
SN - 1557-9654
ER -