{
  "abstract": "We study the quantum query complexity of several basic graph problems. 1) We prove a tight $Θ(n^{3/2})$ bound for finding a maximum-out-degree vertex in an $n$-vertex tournament, resolving a conjecture of Mande, Paraashar, and Saurabh [FSTTCS'23]. 2) Using the same proof idea, we give a direct proof that finding a maximum-degree vertex in an $n$-vertex undirected graph requires $Θ(n^{3/2})$ quantum queries, a bound implicit in the work of Apers and Lee [CCC'21]. Both lower bounds above hold even for computing only the maximum degree or maximum out-degree. 3) We also study Maximum $(k,n-k)$-Cut, which asks for the largest cut with one side of size $k$. For every constant $k\\geq 1$, we give an $O(n^{2-1/(k+1)})$-query quantum algorithm. 4) We establish a reduction to Maximum $(k,n-k)$-Cut involving $k$-Graph Collision, which asks whether the marked vertices of a known graph contain a $k$-clique. For $k\\geq 3$, this yields unconditional lower bounds with exponents strictly larger than $3/2$, showing that this problem is strictly harder than computing maximum degree. Additionally, under a conjecture we propose for $k$-Graph Collision, the reduction yields a lower bound matching our upper bound above.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13906",
  "author_total": 4,
  "authors": [
    {
      "name": "Amin Shiraz Gilani"
    },
    {
      "name": "Nikhil S. Mande"
    },
    {
      "name": "Subhasree Patro"
    },
    {
      "name": "Nithish Raja"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7jg9x91pdj5g0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13906",
  "fulltext_url": null,
  "id": "7jg9x91pdj5g0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13906",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7221530293",
  "pdf_url": "https://arxiv.org/pdf/2610.13906",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7jg9x91pdj5g0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:36Z",
      "native_id": "2610.13906",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13906"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum Query Complexity of Maximum Degree and its Generalizations",
  "title_text": "Quantum Query Complexity of Maximum Degree and its Generalizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:36Z",
  "url": "https://research.draconitic.com/paper/7jg9x91pdj5g0",
  "venue": "arXiv",
  "versions": []
}