Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark
Abstract
Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with kd2/n=12. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula k=2dim R/(A,B) (forcing even k), a 4ℓ m-element symmetry group on generator pairs, an X/Z distance equality dX=dZ, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At n=144 the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: [[144,6,d≥ 15]] exceeds the benchmark distance $12$ (certified d≥ 15), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at kd2/n=11.11 (7.4% below benchmark) while decoding no worse. At n=72, $[[72,14,8]]$ attains kd2/n=12.44---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at ≈ 0.1% pseudo-threshold versus ≈ 0.4% for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.
Related
- Spectral Theory of Semisimple Bivariate Bicycle Codes
- Existence and Characterization of 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
- Sequences of Bivariate Bicycle Codes from Covering Graphs
BibTeX
@misc{lu2026quantum,
title = {{Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark}},
author = {Liangdong Lu and Ruipan Yang and Guanmin Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06572},
eprint = {2609.06572},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06572},
}
RIS
TY - UNPB
TI - Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark
AU - Liangdong Lu
AU - Ruipan Yang
AU - Guanmin Guo
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06572
UR - https://arxiv.org/abs/2609.06572
AB - Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with kd^2/n=12. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula k=2dim R/(A,B) (forcing even k), a 4ℓ m-element symmetry group on generator pairs, an X/Z distance equality d_X=d_Z, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At n=144 the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: [[144,6,d≥ 15]] exceeds the benchmark distance $12$ (certified d≥ 15), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at kd^2/n=11.11 (7.4% below benchmark) while decoding no worse. At n=72, $[[72,14,8]]$ attains kd^2/n=12.44---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at ≈ 0.1% pseudo-threshold versus ≈ 0.4% for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.
ER -