{
  "abstract": "The classical Kolmogorov--Arnold representation theorem states that any continuous multivariate function can be exactly decomposed into a finite composition of univariate continuous functions and addition operations. This foundational result has recently inspired the development of Kolmogorov--Arnold Networks (KANs) in classical machine learning, as well as their extensions into the quantum domain (QKANs). In this paper, we establish two quantum analogues of the Kolmogorov--Arnold representation theorem for continuous unitary-valued maps of several variables within an open $1$-neighbourhood of the identity matrix \\(O_1(\\mathbf{I}) \\subset \\mathcal{U}(n)\\). First, we prove a representation theorem that yields an exact additive decomposition inside the matrix exponent of anti-Hermitian-valued maps. Second, due to the non-commutative nature of quantum operators, we derive a factorised version expressing the target unitary map as a finite sequential product of univariate matrix exponentials. Finally, we provide a concrete topological counterexample based on the lifting property of \\(\\mathcal{SU}(2)\\) to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group \\(\\mathcal{U}(n)\\) without encountering fundamental structural obstructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.03187",
  "author_total": 1,
  "authors": [
    {
      "name": "Sviatoslav V. Dzhenzher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bxcqkx8q1spgc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.03187",
  "fulltext_url": null,
  "id": "bxcqkx8q1spgc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.03187",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7167599183",
  "pdf_url": "https://arxiv.org/pdf/2607.03187",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bxcqkx8q1spgc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2607.03187",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.03187"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps",
  "title_text": "Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/bxcqkx8q1spgc",
  "venue": "arXiv",
  "versions": []
}