{
  "abstract": "It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford $C^{r-1}Z$ gate, for every constant $r\\geq 3$. In particular, we obtain length-$n$ quantum LDPC codes (with constant-weight stabilizers) of polynomial distance $d\\geq n^{(1-ε)/r}$ supporting transversal $C^{r-1}Z$ gates on a close-to-linear number $k\\geq n^{1-ε}$ of disjoint tuples of logical qubits, for arbitrarily small $ε>0$. Our construction is the first with constant-weight stabilizers that obtains $dk\\gg n$, and as a consequence achieves arbitrarily small magic state overhead exponent $γ=\\log(n/k)/\\log(d)>0$. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many $k=Ω(n)$ logical $C^{r-1}Z$ gates, though with stabilizer weight and physical circuit depth $n^ε$. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal $C^{r-1}Z$. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27801",
  "author_total": 3,
  "authors": [
    {
      "name": "Louis Golowich"
    },
    {
      "name": "Itzhak Tamo"
    },
    {
      "name": "Guanyu Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a5yd8mbf40f7q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27801",
  "fulltext_url": null,
  "id": "a5yd8mbf40f7q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27801",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214163733",
  "pdf_url": "https://arxiv.org/pdf/2609.27801",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a5yd8mbf40f7q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:56Z",
      "native_id": "2609.27801",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27801"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Improved Transversal Non-Clifford Gates from Cup Products",
  "title_text": "Improved Transversal Non-Clifford Gates from Cup Products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/a5yd8mbf40f7q",
  "venue": "arXiv",
  "versions": []
}