{
  "abstract": "The Fourier hierarchy $\\mathrm{FH}_0\\subseteq\\mathrm{FH}_1\\subseteq\\mathrm{FH}_2\\subseteq\\cdots$, introduced by Shi (TCS 2005), measures a quantum computation by the number of Hadamard layers it uses. Between two layers the circuit may permute basis states and attach phases, but it may not create superposition; the layers are its only source of interference. The first level is exactly $\\mathrm{BPP}$, while the second already solves Simon's problem and, through phase estimation, factors integers. Shi conjectured that every additional layer strictly increases computational power, and asked, as a first step, for oracle separations between consecutive levels. To our knowledge, the question was open at every level $k\\ge2$. We prove that for every constant $k\\ge2$ there is an oracle relative to which $\\mathrm{FH}_k\\subsetneq\\mathrm{FH}_{k+1}$. The separating problem is built from Forrelation (Aaronson and Ambainis, STOC 2015): the level above solves it with a constant number of queries, whereas at level $k$ it stays hard even for circuits making exponentially many queries. This holds for both of the usual ways of giving a circuit access to an oracle, the phase oracle and the standard oracle, which writes its answer into a register. The two are not interchangeable: relative to an oracle, the standard oracle is strictly more powerful at the same number of layers. We also separate the union of all the levels from $\\mathrm{BQP}$ relative to an oracle. The lower bounds rest on a structural property of the hierarchy: the number of Hadamard layers limits how adaptively a circuit can query its oracle. With a phase oracle, a circuit with $k$ layers is reproduced exactly by an algorithm making only $k-1$ rounds of parallel queries, which brings known lower bounds for such algorithms to bear. The standard oracle lets a circuit branch on earlier answers, and that case needs a separate argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11830",
  "author_total": 1,
  "authors": [
    {
      "name": "Atul Mantri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ck1hnfb5w6fn7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11830",
  "fulltext_url": null,
  "id": "ck1hnfb5w6fn7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11830",
  "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": "W7212262002",
  "pdf_url": "https://arxiv.org/pdf/2609.11830",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ck1hnfb5w6fn7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:26Z",
      "native_id": "2609.11830",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11830"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Oracle Separations in the Fourier Hierarchy",
  "title_text": "Oracle Separations in the Fourier Hierarchy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:26Z",
  "url": "https://research.draconitic.com/paper/ck1hnfb5w6fn7",
  "venue": "arXiv",
  "versions": []
}