{
  "abstract": "The Clifford hierarchy $\\mathsf{C}_1 \\subset \\mathsf{C}_2 \\subset \\cdots$ was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about $\\mathsf{C}_k$ for $k > 3$. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension $d$ or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in $\\cup_k \\mathsf{C}_k$ is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case $d=2$, $k=3$ (arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of) $U \\in \\mathsf{C}_3$ on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group $Γ_1(U)$ generated by $U \\mathsf{P} U^\\dagger$ and beyond, we prove the conjecture for $k \\leq 4$ and any prime dimension $d$. Our proof centers on conjugation groups $Γ_1(U), Γ_2(U), \\ldots$ of $U \\in \\mathsf{C}_k$, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38751",
  "author_total": 3,
  "authors": [
    {
      "name": "Maxwell Marcus"
    },
    {
      "name": "Sathyawageeswar Subramanian"
    },
    {
      "name": "Marcel Dall'Agnol"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2b1mf4dr7pn8d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38751",
  "fulltext_url": null,
  "id": "2b1mf4dr7pn8d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38751",
  "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": "W7215081809",
  "pdf_url": "https://arxiv.org/pdf/2609.38751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2b1mf4dr7pn8d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38751"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The Generalized Semi-Clifford Conjecture Holds at Level 4",
  "title_text": "The Generalized Semi-Clifford Conjecture Holds at Level 4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/2b1mf4dr7pn8d",
  "venue": "arXiv",
  "versions": []
}