{
  "abstract": "We study semi-Clifford and generalized semi-Clifford structures in the Clifford hierarchy as a function of hierarchy level $k$ and qubit number $n$. We show that $\\mathcal{C}_k(n)$ is semi-Clifford if and only if $k\\le2$, $n\\le2$, or $k=3$ and $n\\le6$. We also establish that $\\mathcal{C}_k(n)$ is generalized semi-Clifford when $k\\le4$ or $n\\le3$ and construct counterexamples when $k,n \\geq 5$. For $n=4$ and $k\\ge5$, we show that every gate defined over $\\mathbb{Q}(i)$ up to a scalar is generalized semi-Clifford, reduce the general case to a single level, and propose a strategy to prove our conjecture that $\\mathcal{C}_k(n)$ is generalized semi-Clifford if and only if $k\\le4$ or $n\\le4$. Lastly, we go beyond this structure by introducing $n_c$-generalized semi-Clifford gates, which we show are exactly the gates of the form $\\sum_x|π(x)\\rangle\\langle x|\\otimes V_x$ up to Clifford gates on both sides, where $π$ permutes the computational basis states of $n_c$ control qubits. We conjecture that for $k\\ge3$ every gate in $\\mathcal{C}_k(n)$ has this form for some $n_c\\ge1$ with every $V_x\\in\\mathcal{C}_{k-3}(n-n_c)$, refining the conjecture of Zeng, Chen, and Chuang.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06402",
  "author_total": 2,
  "authors": [
    {
      "name": "Rongbiao Thomas Wang"
    },
    {
      "name": "Bobby Zixuan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c8xxz9tqd0jx8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06402",
  "fulltext_url": null,
  "id": "c8xxz9tqd0jx8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06402",
  "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": "W7220708286",
  "pdf_url": "https://arxiv.org/pdf/2610.06402",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c8xxz9tqd0jx8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.06402",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06402"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "When is the Clifford hierarchy generalized semi-Clifford?",
  "title_text": "When is the Clifford hierarchy generalized semi-Clifford?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/c8xxz9tqd0jx8",
  "venue": "arXiv",
  "versions": []
}