{
  "abstract": "The constructive diagonal Ramsey problem asks, given adjacency-oracle access to an $N$-vertex graph, for a clique or independent set of the order guaranteed by Ramsey's theorem. We give a bounded-error quantum algorithm that, for every $K\\ge2$ and $N\\ge4^{K-1}$, finds and verifies a homogeneous $K$-set using $O\\!\\left(2^K K\\log\\frac Kη\\right)$ edge queries with failure probability at most $η$. At the Ramsey scale $N=2^n$, this yields a homogeneous set of order $\\lfloor n/2\\rfloor+1$ using $O(\\sqrt N\\log N\\log(\\log N/η))$ queries, improving on the $O(N)$ queries of the explicit classical recursion and giving, to our knowledge, the first sublinear worst-case algorithm for the Ramsey relation. We also derive an $Ω(N^{1/12})$ quantum lower bound by a reduction from collision finding. The algorithm runs the constructive recursion over implicit candidate sets. Each set is represented by a short conjunction of adjacency constraints and sampled using capped unknown-solution quantum search, and a scale-aware concentration schedule balances estimation accuracy against the increasing cost of sampling deeper sets. We complement the upper bound with an $Ω(N^{1-1/\\sqrt2})$ randomized lower bound, transported from the random-Painter analysis of online Ramsey numbers, which holds on the uniform distribution $G(N,1/2)$. On that distribution a greedy quantum search uses only $\\widetilde O(N^{1/4})$ queries, giving a provable polynomial quantum speedup for Ramsey search on random graphs. We also give an estimation-free size-biased recursion and extend it to every fixed number of edge colours.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05812",
  "author_total": 1,
  "authors": [
    {
      "name": "Cheng Xin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26fsb9tcfn6nk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05812",
  "fulltext_url": null,
  "id": "26fsb9tcfn6nk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05812",
  "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.DS"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7212081776",
  "pdf_url": "https://arxiv.org/pdf/2609.05812",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/26fsb9tcfn6nk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2609.05812",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05812"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum Query Algorithms for the Constructive Diagonal Ramsey Theorem",
  "title_text": "Quantum Query Algorithms for the Constructive Diagonal Ramsey Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:48Z",
  "url": "https://research.draconitic.com/paper/26fsb9tcfn6nk",
  "venue": "arXiv",
  "versions": []
}