On Constructing and Decoding Quantum Triorthogonal Codes
Abstract
Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical T gates. As a non-Clifford operation, the T gate enables universal fault-tolerant quantum computation when combined with Clifford operations. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate) that are not necessarily constructed from triply even binary linear codes. By imposing automorphism-group constraints, the formulation also recovers the known [[49,1,5]] and [[95,1,7]] triorthogonal codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding (AED) and guessing random additive-noise decoding (GRAND) are promising options.
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BibTeX
@misc{baldelli2026constructing,
title = {{On Constructing and Decoding Quantum Triorthogonal Codes}},
author = {Alessio Baldelli and Olai Å. Mostad and Hsuan-Yin Lin and Eirik Rosnes and Massimo Battaglioni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.24519},
eprint = {2605.24519},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.24519},
}
RIS
TY - UNPB
TI - On Constructing and Decoding Quantum Triorthogonal Codes
AU - Alessio Baldelli
AU - Olai Å. Mostad
AU - Hsuan-Yin Lin
AU - Eirik Rosnes
AU - Massimo Battaglioni
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2605.24519
UR - https://arxiv.org/abs/2605.24519
AB - Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical T gates. As a non-Clifford operation, the T gate enables universal fault-tolerant quantum computation when combined with Clifford operations. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate) that are not necessarily constructed from triply even binary linear codes. By imposing automorphism-group constraints, the formulation also recovers the known [[49,1,5]] and [[95,1,7]] triorthogonal codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding (AED) and guessing random additive-noise decoding (GRAND) are promising options.
ER -