{
  "abstract": "I present a short approach to discretizing continuous unitary evolution. It gives a correspondence in both directions between continuous unitary evolutions and products of unitaries in finite dimensions, and I use it to reduce discrete adiabatic theorems to continuous ones. At fine steps, holding truncated samples of a regulated Hamiltonian on a suitable finite partition gives an approximating product. Conversely, rounding the clock identifies a product of $T$ unit-step samples with a continuous evolution, up to two half-step phases. I bound its operator-norm distance from the original evolution by $O(1/T)$ through cancellation within each time step. This assumes bounded first and second derivatives, bounded spectral width, and eigenvalue differences bounded away from nonzero multiples of $2π$, uniformly in $T$. For a unitary walk with these derivative bounds, a fixed common free spectral arc supplies a Hermitian logarithm path satisfying the comparison hypotheses. Continuous simulation guarantees then transfer to the phase-corrected product, and continuous adiabatic bounds to the product itself, with an explicit additive $O(1/T)$ error and each adiabatic theorem's hypotheses retained. For positive definite quantum linear systems of condition number $κ$, this comparison and continuous adiabatic theory give a rescaled qubitized walk with $O(κ/δ)$ steps for leakage $δ$, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06541",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Jarret"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4vw24eph343n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06541",
  "fulltext_url": null,
  "id": "f4vw24eph343n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06541",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7220674974",
  "pdf_url": "https://arxiv.org/pdf/2610.06541",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4vw24eph343n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.06541",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06541"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Products of unitaries as continuous evolutions at finite precision",
  "title_text": "Products of unitaries as continuous evolutions at finite precision",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/f4vw24eph343n",
  "venue": "arXiv",
  "versions": []
}