{
  "abstract": "Kac's walk is a classical local random walk whose action on a single real unit vector in dimension $d$ mixes in total variation in $Θ(d\\log d)$ sequential steps~\\cite{PS17}. Lu, Qin, Song, Yao, and Zhao introduced a parallel version of Kac's walk that mixes a single quantum state in $O(\\log d)$ rounds~\\cite{LQSY+26}. After discretizing the randomness and replacing it by suitable pseudorandom primitives, this parallel walk gives rise to pseudorandom state scramblers, and was subsequently shown to yield pseudorandom unitaries~\\cite{LQSY+25}. We study what happens when the parallel Kac's walk acts simultaneously on $k$ orthonormal quantum states. We prove that, for any $1\\leq k < d$, after $O\\!\\left((k+\\log d)\\log(d/\\varepsilon)\\right)$ steps, the joint distribution of the $k$ output states is $\\varepsilon$-close, in both Wasserstein and total variation distance, to that obtained by applying a common Haar-random unitary to the same inputs. This generalizes the dispersing property of the parallel Kac's walk from a single quantum state to multiple orthonormal quantum states. Equivalently, viewing an ordered collection of $k$ orthonormal states as a point on the complex Stiefel manifold $V_{d,k}=\\{X\\in\\mathbb C^{d\\times k}:X^\\dagger X=I_k\\}$, we show that the parallel Kac's walk mixes rapidly on $V_{d,k}$, with both Wasserstein and total variation mixing times bounded by $O\\!\\left((k+\\log d)\\log(d/\\varepsilon)\\right)$. This extends the Wasserstein mixing result of Pillai, Smith, and Vaikuntanathan for the standard Kac's walk on real Stiefel manifolds~\\cite{PSV26}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37999",
  "author_total": 5,
  "authors": [
    {
      "name": "Qian Chen"
    },
    {
      "name": "Minglong Qin"
    },
    {
      "name": "Fang Song"
    },
    {
      "name": "Penghui Yao"
    },
    {
      "name": "Mingnan Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f44xsbe0xdyj1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37999",
  "fulltext_url": null,
  "id": "f44xsbe0xdyj1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37999",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214952731",
  "pdf_url": "https://arxiv.org/pdf/2609.37999",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f44xsbe0xdyj1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2609.37999",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37999"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Rapid Mixing of Parallel Kac's Walk: From Spheres to Stiefel Manifolds",
  "title_text": "Rapid Mixing of Parallel Kac's Walk: From Spheres to Stiefel Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:31Z",
  "url": "https://research.draconitic.com/paper/f44xsbe0xdyj1",
  "venue": "arXiv",
  "versions": []
}