{
  "abstract": "An important question in quantum information theory asks whether every gate in the Clifford hierarchy is semi-Clifford or not, as such gates admit resource-efficient implementations via gate teleportation. Extending recent results for prime dimensions~\\cite{silva_clifford_2025}, we prove that every hierarchy gate on a single qudit of dimension \\(d\\) is semi-Clifford, if and only if \\(d\\) is square-free. In composite dimensions, \\(\\mathbb{Z}_d^2\\) is a symplectic module rather than a vector space, motivating a distinction between four types of gates. Semi-Clifford gates ($\\mathcal{SC}$) are the ones that become diagonal under left and right multiplication by Clifford gates, while a broader class $\\mathcal{N}$ admits a permutation diagonal form, under left and right multiplication by Cliffords. Lagrangian semi-Clifford gates ($\\mathcal{LSC}$) conjugate some maximal abelian Pauli subgroup to another, whereas generalized semi-Clifford gates ($\\mathcal{GSC}$) map some maximal abelian Pauli subalgebra to another under conjugation. In square-free dimensions, we have \\(\\mathcal{SC}=\\mathcal{LSC}\\) and \\(\\mathcal{N}=\\mathcal{GSC}\\), but when \\(d\\) is not square-free, these equivalences can fail, as Lagrangian submodules can become non-free. We also show that every third-level gate of the one-qudit Clifford hierarchy is a generalized semi-Clifford in arbitrary dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06836",
  "author_total": 2,
  "authors": [
    {
      "name": "Yifei Qi"
    },
    {
      "name": "Rahul Sarkar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fkwpzyfeg9fhk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06836",
  "fulltext_url": null,
  "id": "fkwpzyfeg9fhk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06836",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7220741451",
  "pdf_url": "https://arxiv.org/pdf/2610.06836",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fkwpzyfeg9fhk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:05Z",
      "native_id": "2610.06836",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06836"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite dimensions",
  "title_text": "Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:05Z",
  "url": "https://research.draconitic.com/paper/fkwpzyfeg9fhk",
  "venue": "arXiv",
  "versions": []
}