{
  "abstract": "Data reuploading---repeatedly encoding classical inputs at multiple layers of a parameterized quantum circuit---is a central mechanism for enhancing the expressivity of quantum neural networks (QNNs). Building on the Fourier-theoretic framework of Schuld et al.~\\cite{schuld2021fourier}, we study the frequency spectrum of the general multivariate, $R$-reupload architecture (QDR). (i) We make explicit that the accessible frequencies form the set $K_R=S^{(R)}-S^{(R)}$, which equals the $R$-fold sumset $T^{(R)}$ of the difference set $T=S-S$ of the encoding spectrum, and we characterise it exactly: $K_R=\\{k\\in\\mathcal L:\\ \\ell_T(k)\\le R\\}$, where $\\mathcal L=\\mathrm{span}_{\\mathbb Z}(T)$ is an integer lattice and $\\ell_T$ is word length in $T$. (ii) We show that the \\emph{directions} of the spectrum stabilise already at $R=1$ (the real span of $K_R$ equals that of $\\mathcal L$), whereas the spectrum itself never saturates: the maximal frequency grows linearly in $R$ and the number of frequencies grows polynomially, as $Θ(R^{m})$ with $m=\\mathrm{rank}\\,\\mathcal L$. (iii) Using the spectrum size, we give a Rademacher-complexity bound on the capacity of QDRs of order $\\sqrt{|K_R|/n}$. (iv) We report experiments on function approximation and classification with train/test splits, several seeds and parameter-matched classical baselines. The simulated spectra match the theory exactly. A single-qubit QDR beats a matched MLP on several smooth or periodic univariate targets and on the spiral task, but it is clearly worse on multivariate regression ($d\\ge4$), where it is highly sensitive to the input-encoding scale; it beats the MLP only on equations with $d\\le3$ inputs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09947",
  "author_total": 2,
  "authors": [
    {
      "name": "Lord Sen"
    },
    {
      "name": "Shyamapada Mukherjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/64rzd30766zkj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09947",
  "fulltext_url": null,
  "id": "64rzd30766zkj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09947",
  "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": "W7220927014",
  "pdf_url": "https://arxiv.org/pdf/2610.09947",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/64rzd30766zkj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:37Z",
      "native_id": "2610.09947",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09947"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "On the Expressive Power and Capacity of Quantum Data Reuploaders",
  "title_text": "On the Expressive Power and Capacity of Quantum Data Reuploaders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:37Z",
  "url": "https://research.draconitic.com/paper/64rzd30766zkj",
  "venue": "arXiv",
  "versions": []
}