Bicycle Race: small two-block codes on the Pareto frontier
Abstract
Two-block group-algebra codes are among the simplest quantum LDPC codes, and many of them are translation invariant on a twisted torus, which suits neutral-atom arrays and other platforms with movable qubits. We ask how far they go at small sizes when searched systematically across all groups. We use algebraic constraints and screens to design a search pipeline capable of enumerating broad and complete swathes of two-block codes on a laptop. With it we census about 108 two-block codes with up to $320$ qubits at check weights five to nine. Several of these censuses are exhaustive, and support complete statements; for instance, the Gross code is, up to equivalence, the only abelian weight-six two-block code with n ≤ 200 to reach kd2/n=12, no weight-six two-block code with n ≤ 300 beats $[[288,16,16]]$, and the known $[[64,18,8]]$ is the best two-block code by kd2/n across weights five to nine with n ≤ 94. The censuses are also large enough to give insights into where good codes live - they concentrate in a handful of groups; abelian groups give the best codes at small n and non-abelian groups at large n, with the crossover appearing to move out as the check weight grows; and high quality codes become $100$ to $1000$ times rarer per draw from weight six to weights eight and nine. Using the group algebra and the census data to steer a statistical search finds four to fifteen times as many good codes as uniform draws in equal time. Among the codes found are $[[102,20,10]]$ and $[[120,24,10]]$ at check weight nine, which we simulate in noisy syndrome-extraction circuits, and $[[186,20,18]]$, a weight-nine code with every check in a 5×5 window. At check weights seven to nine, our best codes exceed the best published kd2/n with exact distance at the same check weight in most length bands up to $320$ qubits. All codes are available in a public database.
Related
- Towards Block-Level Fault-Tolerant Quantum Simulation on Small High-Rate Non-CSS Codes
- Spectral Theory of Semisimple Bivariate Bicycle Codes
- Bivariate Bicycle Codes over Group Algebras
- Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes
- Metachecks in Bivariate Bicycle Codes: Syndrome Distance, Measurement Faults, and Repair Limits
BibTeX
@misc{lee2026bicycle,
title = {{Bicycle Race: small two-block codes on the Pareto frontier}},
author = {Gideon Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13880},
eprint = {2610.13880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13880},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bicycle Race: small two-block codes on the Pareto frontier
AU - Gideon Lee
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.13880
UR - https://arxiv.org/abs/2610.13880
AB - Two-block group-algebra codes are among the simplest quantum LDPC codes, and many of them are translation invariant on a twisted torus, which suits neutral-atom arrays and other platforms with movable qubits. We ask how far they go at small sizes when searched systematically across all groups. We use algebraic constraints and screens to design a search pipeline capable of enumerating broad and complete swathes of two-block codes on a laptop. With it we census about 10^8 two-block codes with up to $320$ qubits at check weights five to nine. Several of these censuses are exhaustive, and support complete statements; for instance, the Gross code is, up to equivalence, the only abelian weight-six two-block code with n ≤ 200 to reach kd^2/n=12, no weight-six two-block code with n ≤ 300 beats $[[288,16,16]]$, and the known $[[64,18,8]]$ is the best two-block code by kd^2/n across weights five to nine with n ≤ 94. The censuses are also large enough to give insights into where good codes live - they concentrate in a handful of groups; abelian groups give the best codes at small n and non-abelian groups at large n, with the crossover appearing to move out as the check weight grows; and high quality codes become $100$ to $1000$ times rarer per draw from weight six to weights eight and nine. Using the group algebra and the census data to steer a statistical search finds four to fifteen times as many good codes as uniform draws in equal time. Among the codes found are $[[102,20,10]]$ and $[[120,24,10]]$ at check weight nine, which we simulate in noisy syndrome-extraction circuits, and $[[186,20,18]]$, a weight-nine code with every check in a 5×5 window. At check weights seven to nine, our best codes exceed the best published kd^2/n with exact distance at the same check weight in most length bands up to $320$ qubits. All codes are available in a public database.
ER -