{
  "abstract": "Triangle finding is a fundamental problem in graph theory that has played a central role in the development of quantum algorithmic techniques. For $N$-vertex graphs given by their adjacency matrices, a long line of work has improved the quantum query upper bound from $O(N^{3/2})$ to $O(N^{5/4})$. However, the best known lower bound has remained the elementary $Ω(N)$ bound obtained from unstructured search. We prove the first superlinear quantum query lower bound for triangle finding, showing that $Ω(N^{13/12})$ queries are necessary. Our result also implies the same lower bound for triangle detection and for detecting every fixed connected non-bipartite graph. The proof builds on the compressed oracle approach of Zhandry and the transfer approaches of Belovs. It compares the computation on graphs with a planted triangle to the computation on random graphs conditioned to be triangle-free, through a map on the input registers that almost commutes with every query. We average over local changes to the input while preserving its distribution, through a procedure known as conditional resampling. Our main technical contribution is a set of estimates that let us control the query error under this averaging, despite the dependence introduced by conditioning a dense graph to be triangle-free.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06751",
  "author_total": 3,
  "authors": [
    {
      "name": "Arjan Cornelissen"
    },
    {
      "name": "Amin Shiraz Gilani"
    },
    {
      "name": "Xingyu Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/anq9xmd5hdgd5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06751",
  "fulltext_url": null,
  "id": "anq9xmd5hdgd5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06751",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/anq9xmd5hdgd5.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:20:56Z",
      "native_id": "2611.06751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06751"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "A Quantum Lower Bound for Triangle Finding",
  "title_text": "A Quantum Lower Bound for Triangle Finding",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:21Z",
  "url": "https://research.draconitic.com/paper/anq9xmd5hdgd5",
  "venue": "arXiv",
  "versions": []
}