{
  "abstract": "The minimal sampling overhead of the quadratic collective phase $e^{-iγK^2}$ over one layer of single-qubit rotations is decided by the continued fraction of $γ/π$. Here $K$ is a collective observable with spectrum $\\{0,\\dots,n\\}$, such as the permutation-symmetric Hamming weight. This gate is the cardinality-penalty layer of constrained optimization, the one-axis-twisting gate of spin squeezing and the Kerr phase of a bosonic mode. Instead of compiling it to two-qubit gates, we sample it as an LCU over such layers at overhead $Γ$. Our results give an optimality theory for this overhead. Ancilla-free sampling reproduces the target's outcome probabilities up to the minimal $Γ$, and no smaller factor works for every input; for permutation-symmetric $K$, independent per-qubit angles or one layer of arbitrary single-qubit gates give no further reduction. Rational angles $πp/q$ in lowest terms cost exactly $q$ once $n\\ge2q-2$, attained uniquely by a uniform $q$-point sampler; the cost is $Θ(n)$ if and only if $γ/π$ is badly approximable, and bounded in $n$ if and only if it is rational. A second result shows what the symmetry buys: without it, the squaring phase of Fourier arithmetic on $m$ qubits costs at least $2^{0.557m-O(1)}$ over single-qubit gates, beyond the $2^{m/2}$ of every cut bound, by a new certificate-lifting argument. For constructions known only through upper bounds, it gives the optimum: the Fourier-based LCU of Carrera Vazquez, Egger and Woerner is, at fixed $n$, optimal up to a constant at typical angles and beaten by $(n+1)/q$ at rational ones with an $n$-independent sampler; the Kerr decomposition of Upreti, Quesada and Chabaud is the unique optimum over phase shifters at rational parameters without photon-number cutoff; and it identifies the rational couplings where one-axis twisting admits optimal cat decompositions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03595",
  "author_total": 4,
  "authors": [
    {
      "name": "Zhihui Wang"
    },
    {
      "name": "Sujit Roy"
    },
    {
      "name": "Manil Maskey"
    },
    {
      "name": "Rahul Ramachandran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ewg9359xmm3ys.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03595",
  "fulltext_url": null,
  "id": "ewg9359xmm3ys",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03595",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7220453331",
  "pdf_url": "https://arxiv.org/pdf/2610.03595",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ewg9359xmm3ys.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.03595",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03595"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy",
  "title_text": "The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/ewg9359xmm3ys",
  "venue": "arXiv",
  "versions": []
}