{
  "abstract": "Angle-encoded variational quantum circuits implement truncated Fourier series. For one-dimensional inputs, a single-qubit re-uploading (serial) circuit represents every function that a multi-qubit circuit with the same encoding budget $E$ can; we find that this exchange of width for depth is not free in first-order training. Because its state lives on the Bloch sphere, all parameters of the serial circuit move the state, at each input, along one complex direction, which we call \\emph{phase-locking}. We prove that this costs no Fourier direction but leaves no direction to spare: however many parameters it has, the serial circuit has at most $2E+2$ tangent directions, one more than its $2E+1$ Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least $6E+6$. The cost in training is measured against this control, as the number of runs out of 100 that reach the optimum within a fixed step budget. On an exact loss with flat-spectrum targets the gap grows with $E$; at $E=32$ the serial circuit converges in 3, 53 and 83 runs under gradient descent, Adam and Levenberg--Marquardt, the control in 99, 100 and 100. On three real-world series, at matched parameter counts, the serial circuit reaches the least-squares optimum in at most 9 runs under Adam and in 32 to 93 under Levenberg--Marquardt, the control in at least 98 and in all 100; its held-out error is up to about $1.5$ times that of the control under Adam, but no larger in the typical run under Levenberg--Marquardt.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21778",
  "author_total": 8,
  "authors": [
    {
      "name": "Michael Poppel"
    },
    {
      "name": "David Bucher"
    },
    {
      "name": "Maximilian Zorn"
    },
    {
      "name": "Markus Baumann"
    },
    {
      "name": "Sebastian Wölckert"
    },
    {
      "name": "Claudia Linnhoff-Popien"
    },
    {
      "name": "Philipp Altmann"
    },
    {
      "name": "Jonas Stein"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99j8b46zkn745.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21778",
  "fulltext_url": null,
  "id": "99j8b46zkn745",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21778",
  "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": "W7222648342",
  "pdf_url": "https://arxiv.org/pdf/2610.21778",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99j8b46zkn745.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:50Z",
      "native_id": "2610.21778",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21778"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits",
  "title_text": "Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:50Z",
  "url": "https://research.draconitic.com/paper/99j8b46zkn745",
  "venue": "arXiv",
  "versions": []
}