Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes
Abstract
We establish that transversal diagonal gates on exact qubit codes of distance at least two can be exhaustively classified within a finite search space. Although code-preserving physical gates can have continuous phases, any projective logical action induced by a transversal diagonal gate can always be realized by a physical gate of finite order. Crucially, we prove that this physical order has an upper bound determined solely by the code parameters. This order bound enables the subset-sum-linear-programming (SSLP) framework introduced in arXiv:2504.20847 to exhaustively enumerate all candidate discrete symmetries, after which the Knill-Laflamme equations determine the remaining continuous code parameters. Applying this method to the nontrivial diagonal sector of exact $((6,2,3))$ codes, SSLP yields $66$ candidate symmetry classes, of which exactly $15$ are realizable, supporting full projective diagonal logical cyclic groups of orders $2$, $3$, and $5$. In particular, for order $5$, the code is unique up to local unitaries and qubit permutations, equivalent to the code constructed in arXiv:2504.20847. This code also attains the minimum length n=6 among exact ((n, 2, ≥ 3)) codes whose full projective transversal group cannot be conjugated into the projective Clifford group.
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BibTeX
@misc{he2026finite,
title = {{Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes}},
author = {Xi He and Chao Zhang and Shilin Huang and Bei Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29955},
eprint = {2610.29955},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29955},
}
RIS
TY - UNPB
TI - Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes
AU - Xi He
AU - Chao Zhang
AU - Shilin Huang
AU - Bei Zeng
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29955
UR - https://arxiv.org/abs/2610.29955
AB - We establish that transversal diagonal gates on exact qubit codes of distance at least two can be exhaustively classified within a finite search space. Although code-preserving physical gates can have continuous phases, any projective logical action induced by a transversal diagonal gate can always be realized by a physical gate of finite order. Crucially, we prove that this physical order has an upper bound determined solely by the code parameters. This order bound enables the subset-sum-linear-programming (SSLP) framework introduced in arXiv:2504.20847 to exhaustively enumerate all candidate discrete symmetries, after which the Knill-Laflamme equations determine the remaining continuous code parameters. Applying this method to the nontrivial diagonal sector of exact $((6,2,3))$ codes, SSLP yields $66$ candidate symmetry classes, of which exactly $15$ are realizable, supporting full projective diagonal logical cyclic groups of orders $2$, $3$, and $5$. In particular, for order $5$, the code is unique up to local unitaries and qubit permutations, equivalent to the code constructed in arXiv:2504.20847. This code also attains the minimum length n=6 among exact ((n, 2, ≥ 3)) codes whose full projective transversal group cannot be conjugated into the projective Clifford group.
ER -