{
  "abstract": "We study variational quantum distribution learning through a hierarchy of Walsh--Fourier approximations on the Boolean cube. At each level, a selected set of target Fourier coefficients defines a spectral truncation, which is projected onto the probability simplex and used as the target of a quantum circuit Born machine. Parameters learned at one level initialize the next through a warm-start map. We prove an end-to-end expected learning guarantee where the approximation term is determined by the omitted Fourier mass, while a normalized unbiased estimator yields an explicit statistical bound for empirical truncations. We then instantiate the abstract discrepancy conditions for total variation distance and relate the resulting distributional error to quantum-state fidelity. The total-variation specialization incurs the explicit factor $2^{n-1}$ under our normalized $\\ell_2$ convention and is therefore informative only for sufficiently concentrated Fourier tails. The framework does not establish global trainability or eliminate barren plateaus; rather, it identifies the conditions under which low-to-high spectral training admits a approximation--estimation--optimization analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06307",
  "author_total": 3,
  "authors": [
    {
      "name": "Taha Hoseinpour Asli"
    },
    {
      "name": "Sajjad Hashemian"
    },
    {
      "name": "Ebrahim Ardeshir-Larijani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0bj52yx6v6521.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06307",
  "fulltext_url": null,
  "id": "0bj52yx6v6521",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06307",
  "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.LG"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7212042082",
  "pdf_url": "https://arxiv.org/pdf/2609.06307",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0bj52yx6v6521.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2609.06307",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06307"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Hierarchical Fourier Approximation for Variational Quantum Distribution Learning",
  "title_text": "Hierarchical Fourier Approximation for Variational Quantum Distribution Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:48Z",
  "url": "https://research.draconitic.com/paper/0bj52yx6v6521",
  "venue": "arXiv",
  "versions": []
}