{
  "abstract": "A major open problem in quantum complexity is the question of unitary synthesis---namely, is it possible to efficiently implement any unitary, given the ability to evaluate any classical function in superposition? Inspired by a recent work by Brakerski and Yuen (CRYPTO 2026), we propose an ``average-case'' unitary synthesis problem, which asks whether it is possible to efficiently implement a Haar random unitary given access to a random Boolean function. We then demonstrate a superpolynomial query lower bound, establishing that unitary synthesis in this model is impossible. Along with our main result, we settle in the negative a question Brakerski and Yuen pose, of whether scalable pseudorandom unitaries (PRUs) can be implemented in the ROM-PRU model. In particular, we show that in the ROM-PRU model, no $Ω(N^{1+γ})$ unitary design on $n$ qubits, where $N=2^n$, can be efficiently achieved for any $γ> 0$. On the other hand, we construct an $O(N)$-design in $\\text{poly}(n)$ queries in the same model, surpassing the previously best known results which constructed $O(\\sqrt{N})$-designs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06700",
  "author_total": 3,
  "authors": [
    {
      "name": "Andrew Huang"
    },
    {
      "name": "Akshar Ramkumar"
    },
    {
      "name": "John Wright"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9djmx2k943w7g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06700",
  "fulltext_url": null,
  "id": "9djmx2k943w7g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06700",
  "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": "W7220684430",
  "pdf_url": "https://arxiv.org/pdf/2610.06700",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9djmx2k943w7g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.06700",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06700"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Random-Oracle Unitary Synthesis is Impossible",
  "title_text": "Random-Oracle Unitary Synthesis is Impossible",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/9djmx2k943w7g",
  "venue": "arXiv",
  "versions": []
}