{
  "abstract": "Encrypted training relies on keeping server-side updates low-degree. This constraint traditionally excludes models whose weights inhabit a compact Lie group (notably variational quantum circuits, where every trainable weight is an $\\mathrm{SU(2)}$ rotation). Expressed in Euler angles or discrete alphabets, these updates appear transcendental, historically demanding prohibitive costs: one client--server round per gate, or upwards of $25{,}000$ operations per weight. This penalty is strictly an artefact of coordinates. In the unit-quaternion (spin) chart, group composition is exactly bilinear (degree two, with coefficients in $\\{-1,0,+1\\}$). Consequently, encrypted rotation updates cost one multiplicative level and federated averaging costs zero in any levelled homomorphic scheme, completely eliminating bootstrapping. This implementation-independent algebraic property is confirmed across two cryptographic backends, introducing only $0.0$ and $-2.0\\times10^{-12}$ rad of aggregation error. Leveraging this reduction yields a non-interactive protocol for encrypted federated training of hybrid quantum--classical networks. It includes correctness proofs for aggregation and sign handling, plus a compilation lemma proving parameterised entanglers add only constant-factor overhead without altering the depth class. Empirically, a paired five-seed study confirms zero measurable utility tax ($Δ=+9\\times10^{-6}$ MSE, $p=0.92$), and a noise-budget ablation falsifies the hypothesis that encryption noise regularises. These convergence trends replicate across datasets and scale to $20$ clients. Finally, hardware validation on a $156$-qubit processor achieves $0.9918$ fidelity against a $0.99957$ unencrypted control.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30581",
  "author_total": 5,
  "authors": [
    {
      "name": "Marcel Mordarski"
    },
    {
      "name": "Nathan Mani"
    },
    {
      "name": "Arshad Patel"
    },
    {
      "name": "William Knottenbelt"
    },
    {
      "name": "Roberto Bondesan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/47cmq2n13t989.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30581",
  "fulltext_url": null,
  "id": "47cmq2n13t989",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30581",
  "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.CR"
    ],
    [
      "arxiv",
      "cs.DC"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.30581",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/47cmq2n13t989.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:52Z",
      "native_id": "2609.30581",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30581"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Encryptability As a Coordinate Choice: Depth-One Homomorphic Federated Learning of Quantum Neural Networks",
  "title_text": "Encryptability As a Coordinate Choice: Depth-One Homomorphic Federated Learning of Quantum Neural Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:52Z",
  "url": "https://research.draconitic.com/paper/47cmq2n13t989",
  "venue": "arXiv",
  "versions": []
}