{
  "abstract": "We achieve the first constant-depth circuit for arbitrary single-qubit gate synthesis. Unlike prior approaches, the construction is fully unitary and requires no pre-supplied catalyst. For any constant $δ>0$, it $\\varepsilon$-approximates an arbitrary single-qubit gate using $O(\\log^{1+δ}(1/\\varepsilon))$ clean ancillae, Hadamard and $T$ single-qubit gates, $O(\\log(1/\\varepsilon))$-width generalized Toffoli gates, and sublogarithmic-width Fan-Out gates. We further eliminate Fan-Out entirely, showing that Hadamard, $T$, and generalized Toffoli gates alone suffice for constant-depth synthesis. When restricted to the standard bounded-width gate model, our construction has depth $O(\\log\\log(1/\\varepsilon))$, and we prove a matching $Ω(\\log\\log(1/\\varepsilon))$-depth lower bound. Overall, we establish that $Θ(\\log\\log(1/\\varepsilon))$-depth is unavoidable with only bounded-width gates, yet allowing even logarithmic-width multi-qubit gates suffices to achieve constant-depth synthesis. These results also reveal new structure in shallow quantum circuit complexity. We give a depth-preserving real simulation of bounded-error decision computation, showing that every depth-$d$ QAC circuit can be simulated in depth $O(d)$ using only Hadamard, $X$, and generalized Toffoli gates. Thus arbitrary single-qubit rotations and complex amplitudes do not increase the bounded-error decision power of QAC, even at constant depth. In particular, this reduces the long-standing conjecture Parity$\\notin$QAC$^0$ to proving a Parity lower bound against circuits consisting only of Hadamard, $X$, and generalized Toffoli gates. More generally, this real normal form exposes a direct correspondence between the standard shallow-depth quantum circuit hierarchy and a hierarchy of Forrelation circuits with restricted oracle families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34659",
  "author_total": 1,
  "authors": [
    {
      "name": "Francisca Vasconcelos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3gr14djf0h2m1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34659",
  "fulltext_url": null,
  "id": "3gr14djf0h2m1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34659",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214773536",
  "pdf_url": "https://arxiv.org/pdf/2609.34659",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3gr14djf0h2m1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.34659",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34659"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Depth-Optimal Quantum Compilation",
  "title_text": "Depth-Optimal Quantum Compilation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:12Z",
  "url": "https://research.draconitic.com/paper/3gr14djf0h2m1",
  "venue": "arXiv",
  "versions": []
}