{
  "abstract": "The $T$-count is a dominant cost of fault-tolerant Clifford$+T$ computation. We prove that every Hadamard-free Clifford$+T$ circuit with clean ancillas for a layer of $m$ disjoint CCZ gates, the diagonal core of a parallel Toffoli layer, uses at least $6m+1$ $T$ gates. Campbell and Howard gave a circuit with $6m+1$, and to our knowledge this is the first proof that it is optimal for general $m$. The proof combines Campbell and Howard's self-orthogonality of the columns of the parity matrix with a rank (flattening) bound, sharpened by one. More generally, the argument gives a lower bound on the $T$-count of every diagonal gate in the third level of the Clifford hierarchy, the isotropy floor, computable by Gaussian elimination. It is never below the stabilizer nullity $ν$ and reaches $2ν+1$ on non-Clifford gates whose phase is purely cubic. No bound that also holds for circuits with measurement and feedforward can do this, since such circuits implement CCZ ($ν=3$) with four $T$ gates and the layer with $4m$, where the floor gives seven and $6m+1$. The floor also recovers Campbell and Howard's count $4m+3$ for a fan-out of Toffolis. On the output of PyZX's TODD-based optimizer, relative to its Hadamard placement, it certifies $193$ of $311$ phase-polynomial blocks $T$-optimal once repeated parities are merged (at the best of five optimizer seeds), against $113$ for nullity, including $37$ of the $72$ blocks too large for exhaustive search.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01024",
  "author_total": 1,
  "authors": [
    {
      "name": "Arul Rhik Mazumder"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/146etpwhb5kfg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01024",
  "fulltext_url": null,
  "id": "146etpwhb5kfg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01024",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7216067610",
  "pdf_url": "https://arxiv.org/pdf/2610.01024",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/146etpwhb5kfg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:28Z",
      "native_id": "2610.01024",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01024"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Exact $T$-counts of CCZ layers from an isotropy bound",
  "title_text": "Exact T-counts of CCZ layers from an isotropy bound",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:47Z",
  "url": "https://research.draconitic.com/paper/146etpwhb5kfg",
  "venue": "arXiv",
  "versions": []
}