{
  "abstract": "We study a generalization of undirected $st$-connectivity to graphs whose edges carry quantum operations. Let $G=(V,E)$ be an undirected graph on $n$ vertices in which each edge $\\{u,v\\}$ is labeled by a unitary $U_{uv}\\in\\mathbb{C}^{k\\times k}$, with $U_{vu}=U_{uv}^\\dagger$. We assume the labels form a \\emph{flat} connection: the ordered product of labels along any path between a pair of vertices $u$ and $v$ is independent of the path. Equivalently, the connection is pure gauge, i.e., gauge-equivalent to the trivial connection; such graphs are exactly the consistent connection graphs of spectral graph theory and the noiseless instances of group synchronization. Consequently, whenever $s$ and $t$ are connected, transporting a state from $s$ to $t$ defines a unique unitary $U_s(t)$. Given states $|ψ_s\\rangle,|ψ_t\\rangle\\in\\mathbb{C}^k$ and an oracle that returns the neighbours of a vertex while coherently applying the corresponding edge unitaries, the \\emph{$st$-transport problem} is to decide whether $s$ and $t$ are connected and, if so, to estimate the squared overlap between $U_s(t)|ψ_s\\rangle$ and $|ψ_t\\rangle$ to additive error $\\varepsilon$. When $k=1$ and all labels are trivial, this is exactly undirected $st$-connectivity. We give a bounded-error quantum algorithm for $st$-transport that runs in time $\\widetilde{O}(n/\\varepsilon)$ and uses $O(\\log n+\\log k+\\log(1/\\varepsilon))$ space. We do this by designing a transducer and applying a Metropolis-Hastings reweighting to the input graph. We also prove an $Ω(n)$ quantum query lower bound that holds even when $s$ and $t$ are promised to be connected, so for constant $\\varepsilon$ our algorithm is optimal up to polylogarithmic factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40231",
  "author_total": 3,
  "authors": [
    {
      "name": "Stacey Jeffery"
    },
    {
      "name": "Tobias J. Osborne"
    },
    {
      "name": "Galina Pass"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/47efxwdhnbx4w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40231",
  "fulltext_url": null,
  "id": "47efxwdhnbx4w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40231",
  "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": "W7215041099",
  "pdf_url": "https://arxiv.org/pdf/2609.40231",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/47efxwdhnbx4w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40231",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40231"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "A Quantum Algorithm for $st$-Transport on Flat Connection Graphs",
  "title_text": "A Quantum Algorithm for st-Transport on Flat Connection Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/47efxwdhnbx4w",
  "venue": "arXiv",
  "versions": []
}