{
  "abstract": "We study braid representations obtained from character-twisted Burau inputs by the Katz-Long-Moody (KLM) construction, which corresponds to multiplicative middle convolution for KZ-type equations. For $n\\ge2$, let $q=e^{2πia}$, $τ=e^{2πic}$, where $a,c>0$ and $nc+(n+1)a<1$. Known form and signature results give a positive-definite invariant form for $λ=e^{2πil}$, $0<l<1-nc-(n+1)a$. For fixed input, unitary realizations are unique up to simultaneous unitary conjugation. We address two questions. First, we determine spectra and invariant summands of the $M=n(n+1)$-dimensional output. The generator spectrum recovers $(q,τ)$. The output decomposes into two trivial and two reduced Burau summands, and an $n(n-1)$-dimensional summand isomorphic to Long's reduced construction, generically irreducible for $n=3,4$. If $a$ or $c$ is irrational, every generator is non-Clifford in every finite-dimensional Weyl-Heisenberg frame; if $1,a,c$ are linearly independent over $\\mathbb{Q}$, every generator is entangling for every nontrivial bipartition. The full braid image is not projectively dense. For general positive-definite unitary inputs, irreducibility of the full braid output at a nonexceptional parameter forces the free-group input operators to commute. Second, we construct one orthonormal basis giving each generator one block of size $2(n+1)$, $n-2$ blocks of size two, and identity elsewhere. Each generator is a product of $2n$ rank-one phase reflections and, in a fixed binary encoding, an $O(M)$-length word in one-qubit unitaries and CNOTs, without auxiliary qubits and including direction preparation and its inverse. We give finite-alphabet approximation, conditioning and error bounds, and a matched-resource comparison with a Quantum Shannon reference compiler. The gate bound is exponential in the encoded qubit count and excludes classical preprocessing.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00293",
  "author_total": 1,
  "authors": [
    {
      "name": "Haru Negami"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6hj3xgsggbr2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00293",
  "fulltext_url": null,
  "id": "c6hj3xgsggbr2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00293",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7216468021",
  "pdf_url": "https://arxiv.org/pdf/2610.00293",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6hj3xgsggbr2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.00293",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00293"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum gates from the middle convolution of twisted Burau representations",
  "title_text": "Quantum gates from the middle convolution of twisted Burau representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/c6hj3xgsggbr2",
  "venue": "arXiv",
  "versions": []
}