{
  "abstract": "We establish tight lower bounds on the adaptive and non-adaptive quantum query complexities of Promised $k$-to-one Finding on an exactly $k$-to-one input string $x\\in\\mathbb Z_q^n$, for every fixed $k\\ge2$ and every feasible alphabet size $q\\ge n/k$. The lower bounds are $Ω_k(n^{1/2-1/(2(2^k-1))})$ for unrestricted adaptive algorithms and $Ω_k(n^{1-1/k})$ for one-round non-adaptive algorithms. The former matches the adaptive quantum algorithm of Liu and Zhandry [LZ19], whereas the latter matches classical birthday sampling even when the parallel queries are arbitrarily entangled. We also establish the non-adaptive lower bound $Ω_k(\\sqrt{n/k})$ for $k$-to-one Testing when $q\\ge n$, resolving an open question posed by Koiran, Landes, Portier, and Yao [KLPY10]. Our proofs generate the balanced input from a uniformly random permutation and use compressed permutation oracles to describe the information learned by the algorithm. The adaptive proof bounds the growth of partial collisions through successive queries. The one-round proof instead analyzes the state after the parallel queries directly, bounding the probability that the compressed database contains a full collision. In both proofs, a short verification step accounts for correct outputs that include unrecorded positions. For the decision problem, we also compare one-round states with independent and distinct output labels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01767",
  "author_total": 2,
  "authors": [
    {
      "name": "Pan Peng"
    },
    {
      "name": "Jingyu Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/btyaw8m75976v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01767",
  "fulltext_url": null,
  "id": "btyaw8m75976v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01767",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01767",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/btyaw8m75976v.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.01767",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01767"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum Lower Bounds for Collision Problems",
  "title_text": "Quantum Lower Bounds for Collision Problems",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:18Z",
  "url": "https://research.draconitic.com/paper/btyaw8m75976v",
  "venue": "arXiv",
  "versions": []
}