{
  "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 $((n, 2, \\ge 3))$ codes whose full projective transversal group cannot be conjugated into the projective Clifford group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29955",
  "author_total": 4,
  "authors": [
    {
      "name": "Xi He"
    },
    {
      "name": "Chao Zhang"
    },
    {
      "name": "Shilin Huang"
    },
    {
      "name": "Bei Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5nqjvyqnkvfhb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29955",
  "fulltext_url": null,
  "id": "5nqjvyqnkvfhb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29955",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7223250166",
  "pdf_url": "https://arxiv.org/pdf/2610.29955",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5nqjvyqnkvfhb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:19Z",
      "native_id": "2610.29955",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29955"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes",
  "title_text": "Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/5nqjvyqnkvfhb",
  "venue": "arXiv",
  "versions": []
}