{
  "abstract": "We determine which representation sectors control the slowest convergence of random Pauli rotations and several local random quantum circuits. For $d=2^n$, the continuous Pauli walk has exact all-representation gaps $$Δ_{\\mathsf{SU}(d)}=\\frac{d-8}{8(d-1)}\\quad(n\\ge4),\\qquad Δ_{\\mathsf{PU}(d)}=\\frac{d(d-3)}{8(d^2-1)}\\quad(n\\ge3).$$ The special-unitary gap is attained in $\\bigwedge^8\\mathbb{C}^d$ and is invisible to balanced moments, while the projective gap is already attained in a balanced fourth-moment representation. We obtain exact discrete and orthogonal Pauli analogues, exact finite-Clifford chain/path gaps, and dimension-independent all-representation Haar brickwork lower bounds; on three qubits every weighted Haar-$\\mathsf{SU}(4)$ edge walk reduces to one universal principal-angle constant. For real complete-graph models, the entire ordinary fourth-tensor gap reduces to a quadratic-size occupation operator, is simple for every $n\\ge4$, and has an expansion beginning $$Δ_n^{(4)}=\\frac{10}{9n}-\\frac{88}{105n^2}+O(n^{-3}),$$ with competing branches separated only beyond all algebraic orders. For the two finite real Clifford complete-graph walks, this fourth-tensor mode is the exact all-representation group gap for every $n\\ge144$; in the same range it simultaneously governs every ordinary even tensor degree $k\\ge4$. Thus different representation scopes can have genuinely different slow modes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40164",
  "author_total": 2,
  "authors": [
    {
      "name": "Ziyuan Dong"
    },
    {
      "name": "Xiang Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f32sa8h4mte11.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40164",
  "fulltext_url": null,
  "id": "f32sa8h4mte11",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40164",
  "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.PR"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7215087780",
  "pdf_url": "https://arxiv.org/pdf/2609.40164",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f32sa8h4mte11.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2609.40164",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40164"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Spectral gaps and slow modes of Pauli rotations and random quantum circuits",
  "title_text": "Spectral gaps and slow modes of Pauli rotations and random quantum circuits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/f32sa8h4mte11",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.5281/zenodo.23099545",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}