{
  "abstract": "Whether $\\mathsf{QAC}^0$ can compute $\\mathtt{PARITY}_n$ remains open. Computing $\\mathtt{PARITY}_n$ is equivalent to implementing $\\mathtt{FANOUT}_n$ under $\\mathsf{QAC}^0$ reductions. This raises a more general question: for an arbitrary symmetric Boolean function $f:\\{0,1\\}^n\\to\\{0,1\\}$, what fanout size is necessary and sufficient for computing $f$ in $\\mathsf{QAC}^0$? We show that the answer is exactly the transition radius $ρ(f)$: computing $f$ and implementing $\\mathtt{FANOUT}_{ρ(f)}$ are equivalent under $\\mathsf{QAC}^0$ reductions. In particular, if $ρ(f)\\ge n^δ$ for some constant $δ>0$, then computing $f$ is $\\mathsf{QAC}^0_{\\mathrm{f}}$-complete. Combined with Paturi's theorem, our characterization implies that if $\\mathtt{PARITY}_n \\notin \\mathsf{QAC}^0$, then any Boolean function in $\\mathsf{QAC}^0$ of approximate degree $n^{1/2+Ω(1)}$ must be nonsymmetric.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05153",
  "author_total": 2,
  "authors": [
    {
      "name": "Boyan Xu"
    },
    {
      "name": "Lvzhou Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/98c8xdeh8cwm9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05153",
  "fulltext_url": null,
  "id": "98c8xdeh8cwm9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05153",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7211903978",
  "pdf_url": "https://arxiv.org/pdf/2609.05153",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/98c8xdeh8cwm9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2609.05153",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05153"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Fanout Complexity of Symmetric Boolean Functions in $\\mathsf{QAC}^0$",
  "title_text": "Fanout Complexity of Symmetric Boolean Functions in QAC^0",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:14Z",
  "url": "https://research.draconitic.com/paper/98c8xdeh8cwm9",
  "venue": "arXiv",
  "versions": []
}