{
  "abstract": "We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs $G_1,\\ldots,G_m$ on $n$ labeled vertices, let $\\ell_b$ be the snapshot lifetime of a degree-one bar $b$ of the intersection zigzag. For a probability generating function $ϕ(x)=\\mathbb{E}[x^R]$, the statistic $F_ϕ=\\sum_bϕ(\\ell_b/m)$ includes normalized degree-$r$ total persistence and the mean generalized rank over a uniform time window. An exact identity underlies our algorithm: sample $R$ uniform times; the expected generalized rank between their minimum and maximum equals $F_ϕ$. For graphs that rank is the circuit rank of an intersection graph, so a nonlinear barcode functional becomes an average of edge and component counts, and no barcode is computed. Without spectral-gap, homology-state, or QRAM assumptions, this gives a quantum estimator with additive error $\\varepsilon n$ and $\\widetilde O(\\sqrt{m(K+n)}/\\varepsilon)$ queries when a bound $K\\ge F_ϕ$ is supplied, against $\\widetilde O(m\\min\\{n^2,(K+n)/\\varepsilon^2\\})$ classically, and an adaptive quantum variant with the same instance dependence. These estimators are optimal in two regimes. For every fixed power weight $x^r$, $r\\ge2$, and for the uniform-window mean, the worst-case complexities are $\\widetildeΘ(n\\sqrt m/\\varepsilon)$ quantum and $Θ(n^2m)$ classical. On sparse instances, under an explicit split-leakage promise met by power and binomial weights of logarithmic degree and the promise $F_ϕ\\le K$, they are $\\widetildeΘ(\\sqrt{mK}/\\varepsilon)$ and $\\widetildeΘ(m\\min\\{n^2,K/\\varepsilon^2\\})$. The classical lower bounds hold against fully adaptive algorithms, and fewer than $m$ such statistics cannot determine the positive-lifetime histogram. All bounds concern snapshot access; with an explicit update stream, near-linear full-barcode algorithms are known.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05813",
  "author_total": 1,
  "authors": [
    {
      "name": "Cheng Xin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ffvsme3md838.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05813",
  "fulltext_url": null,
  "id": "7ffvsme3md838",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05813",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7212000034",
  "pdf_url": "https://arxiv.org/pdf/2609.05813",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ffvsme3md838.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2609.05813",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05813"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum Query Complexity of Persistence Statistics in Graph Zigzags",
  "title_text": "Quantum Query Complexity of Persistence Statistics in Graph Zigzags",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:48Z",
  "url": "https://research.draconitic.com/paper/7ffvsme3md838",
  "venue": "arXiv",
  "versions": []
}