{
  "abstract": "Span programs provide a general method for designing quantum query algorithms, and for Boolean coordinate queries their optimum cost agrees with the general adversary bound. We develop a matroidal organization of this optimization that separates dependence data determined by the query problem from the dependence structure chosen by a span program. The resulting source matroid records query dependence, target determination, and separation data, while a program matroid records dependencies among the target and response-labeled input elements of a candidate program. When a query problem admits a source matroid, we use its structure to organize adversary upper and lower bounds. For separating binary query sets in this setting, the global matroid-span-program optimum equals $\\mathrm{Adv}^{\\pm}_{\\mathcal Q}(f)$. For a fixed regular program matroid, projective uniqueness reduces optimization over real representations to positive element weights. We express positive and negative costs as weighted basis ratios and derive exact $1$-, $2$-, and $3$-sum reductions, giving a compositional evaluation of the fixed-weight objective along a Seymour decomposition. For pointed $R_{10}$ closure, exact certificates give $3.9087553<\\mathrm{Adv}^{\\pm}_{\\mathcal Q_R}(f_R)<3.9301$. The natural $R_{10}$ program has optimized cost $5$, while another regular program has cost at most $\\sqrt{87/5}$. Recursive self-composition yields a family on $N$ input bits with randomized query complexity $Ω(N^{0.7324867\\ldots})$, an explicit quantum algorithm using $O(N^{0.6500178\\ldots})$ queries, and $Q(F)=Θ(N^α)$ for $0.6204277\\ldots<α<0.6229063\\ldots$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03914",
  "author_total": 5,
  "authors": [
    {
      "name": "Justin Roy Cox"
    },
    {
      "name": "Neil Epstein"
    },
    {
      "name": "Zhirui Hu"
    },
    {
      "name": "Michael Jarret"
    },
    {
      "name": "Thomas De Mastri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d2y28fp4bbzgk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03914",
  "fulltext_url": null,
  "id": "d2y28fp4bbzgk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03914",
  "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": "W7220661753",
  "pdf_url": "https://arxiv.org/pdf/2610.03914",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d2y28fp4bbzgk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.03914",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03914"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum Query Complexity and Span Programs from Pre-Geometry",
  "title_text": "Quantum Query Complexity and Span Programs from Pre-Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/d2y28fp4bbzgk",
  "venue": "arXiv",
  "versions": []
}