Designing Group-Valued Codes with Full Regular Low-Weight Bases
Abstract
Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width f, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance d and check weight w. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations. First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by $1$ guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an $[[1088,128,22]]$ LP code with f=22 and an $[[1024,256,24]]$ PP code with f ≤ 27, both at w=10. At w=11, our PP constructions attain either d=25 with f ≤ 27 or d=24 with the proved optimum f=25, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.
Related
- A terminal-group and fluorination matrix for designing polar nematic liquid crystals
- A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group
- Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts
- Low-Weight Canonical Logical Bases from Pair-Partition Codes
- High-Rate Quantum Codes with Proven Distance and Low-Weight Measurements
BibTeX
@misc{lee2026designing,
title = {{Designing Group-Valued Codes with Full Regular Low-Weight Bases}},
author = {Jong Yeon Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06820},
eprint = {2610.06820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06820},
}
RIS
TY - UNPB
TI - Designing Group-Valued Codes with Full Regular Low-Weight Bases
AU - Jong Yeon Lee
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06820
UR - https://arxiv.org/abs/2610.06820
AB - Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width f, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance d and check weight w. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations. First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by $1$ guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an $[[1088,128,22]]$ LP code with f=22 and an $[[1024,256,24]]$ PP code with f ≤ 27, both at w=10. At w=11, our PP constructions attain either d=25 with f ≤ 27 or d=24 with the proved optimum f=25, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.
ER -