Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes
Abstract
Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the Z-logical quotient K/S over π½2[x,y]/(xββ1,ymβ1) and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity dZ=min(dπ,dπ). Using the Frobenius structure of the finite group algebra, we prove rπ=rπ=k/2 for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with dπ=10 and dπ=12.
Related
- Metachecks in Bivariate Bicycle Codes: Syndrome Distance, Measurement Faults, and Repair Limits
- Spectral Theory of Semisimple Bivariate Bicycle Codes
- Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark
- fSRD: Fuzzy Spectral Region Decomposition -- Automated Multi Operator Koopman Representations via an Adaptive Spectral Learning Architecture
- The bivariate generating function for cylindric partitions of rank three
BibTeX
@misc{rowshan2026logical,
title = {{Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes}},
author = {Mohammad Rowshan and Simon Devitt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39395},
eprint = {2609.39395},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39395},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes
AU - Mohammad Rowshan
AU - Simon Devitt
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.39395
UR - https://arxiv.org/abs/2609.39395
AB - Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the Z-logical quotient K/S over π½_2[x,y]/(x^ββ1,y^mβ1) and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity d_Z=min(d_π,d_π). Using the Frobenius structure of the finite group algebra, we prove r_π=r_π=k/2 for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with d_π=10 and d_π=12.
ER -