{
  "abstract": "We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $\\mathsf{SU}(2^n)$ on $n$ qubits. (i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $θ\\in [0,2π)$, both uniformly at random, and apply $e^{\\mathrm i θP}$. (ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\\mathsf{SU}(4)$ (including central phases of the parent group) independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$. Importantly, the gap bounds are independent of $n$ and apply for all finite dimensional unitary representations of $\\mathsf{SU}(2^n)$ uniformly. Our results immediately imply that the Random Pauli Rotation and Brickwork Random Unitary Circuit converge to approximate unitary $t$-designs after $\\mathcal{O}(nt)$ steps. As a consequence, we improve implications for the complexity growth of random unitary circuits as a function of the depth. The existing argument can be applied almost verbatim, but previously the depth had to be $\\mathcal{O}(2^{cn})$ for some $c < 1$. Our unconditional constant spectral gap removes this restriction. We prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit. The former implies convergence to approximate orthogonal $t$-designs. In this case, the implied mixing time with respect to a constant additive and multiplicative error is optimal (up to an absolute constant factor) for $t=2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.20919",
  "author_total": 2,
  "authors": [
    {
      "name": "Tim Baer"
    },
    {
      "name": "Jeongwan Haah"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0myjs7fszq1vy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.20919",
  "fulltext_url": null,
  "id": "0myjs7fszq1vy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.20919",
  "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": "W7170930518",
  "pdf_url": "https://arxiv.org/pdf/2607.20919",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0myjs7fszq1vy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:48Z",
      "native_id": "2607.20919",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.20919"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Random unitary circuits with constant spectral gap",
  "title_text": "Random unitary circuits with constant spectral gap",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/0myjs7fszq1vy",
  "venue": "arXiv",
  "versions": []
}