Asymptotically good binary triorthogonal codes and higher-level transversal gates
Abstract
For each level of the Clifford hierarchy from the third level onward, we construct explicit asymptotically good binary CSS codes for which a strongly transversal diagonal rotation, followed by a correction from the preceding level, implements the corresponding logical rotation. We achieve this using algebraic geometry codes over binary extension fields and performing an alphabet reduction that translates orthogonality conditions over the extension field to binary overlap conditions. In particular, we obtain the first asymptotically good family of binary triorthogonal codes. We also give a stronger construction with worse parameters for which the strongly transversal physical rotation implements the desired logical rotation, with no subsequent correction. Finally, adapting an error-correction-based distillation protocol, our binary T-gate families yield direct constant-overhead T-state block distillation from noisy T-state inputs under sufficiently weak input noise and ideal stabilizer operations.
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BibTeX
@misc{sanjose2026asymptotically,
title = {{Asymptotically good binary triorthogonal codes and higher-level transversal gates}},
author = {Rodrigo San-José},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08203},
eprint = {2609.08203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08203},
}
RIS
TY - UNPB
TI - Asymptotically good binary triorthogonal codes and higher-level transversal gates
AU - Rodrigo San-José
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.08203
UR - https://arxiv.org/abs/2609.08203
AB - For each level of the Clifford hierarchy from the third level onward, we construct explicit asymptotically good binary CSS codes for which a strongly transversal diagonal rotation, followed by a correction from the preceding level, implements the corresponding logical rotation. We achieve this using algebraic geometry codes over binary extension fields and performing an alphabet reduction that translates orthogonality conditions over the extension field to binary overlap conditions. In particular, we obtain the first asymptotically good family of binary triorthogonal codes. We also give a stronger construction with worse parameters for which the strongly transversal physical rotation implements the desired logical rotation, with no subsequent correction. Finally, adapting an error-correction-based distillation protocol, our binary T-gate families yield direct constant-overhead T-state block distillation from noisy T-state inputs under sufficiently weak input noise and ideal stabilizer operations.
ER -