{
  "abstract": "Given oracle access to an unknown unitary $U=e^{iH}$ , the fractional query problem asks how many queries are required to implement a noninteger power $U^t=e^{itH}$, $0<t<1$, when the spectrum is separated from the branch cut by a gap $δ$. Quantum singular value transformation gives an upper bound of $O\\!\\left(\\frac{1}δ\\log\\frac{1}{\\varepsilon}\\right)$ queries for approximation error $\\varepsilon$. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any $N$-query circuit to the approximation of $e^{itθ}$ by a trigonometric polynomial with degree bounded by $O(N)$, together with Remez inequality. This allows us to establish the lower bound of $Ω_τ\\!\\left(\\frac{1}δ\\log\\frac{1}{\\varepsilon}\\right)$. Consequently, the optimal query complexity for fractional query problem is $Θ_τ\\!\\left(\\frac{1}δ\\log\\frac{1}{\\varepsilon}\\right)$, showing that the known QSVT construction is asymptotically optimal. We also give an alternative lower bound proof based on constructing a linear functional that annihilates the approximant space, yielding a $Ω_τ\\!\\left(\\log\\frac{1}{\\varepsilon}\\right)$ bound uniform to $δ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01940",
  "author_total": 5,
  "authors": [
    {
      "name": "Anthony Yuezhang Liu"
    },
    {
      "name": "Adam Wesołowski"
    },
    {
      "name": "Jayne Thompson"
    },
    {
      "name": "Mile Gu"
    },
    {
      "name": "Lirandë Pira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3em63yb0az77x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01940",
  "fulltext_url": null,
  "id": "3em63yb0az77x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01940",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7216059686",
  "pdf_url": "https://arxiv.org/pdf/2610.01940",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3em63yb0az77x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.01940",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01940"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal query complexity for fractional quantum evolution",
  "title_text": "Optimal query complexity for fractional quantum evolution",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:03Z",
  "url": "https://research.draconitic.com/paper/3em63yb0az77x",
  "venue": "arXiv",
  "versions": []
}