{
  "abstract": "An $[[n,k,d]]$ stabilizer code is phantom if every in-block logical CNOT gate can be realized by a permutation of its physical qubits. This eliminates the large and complicated physical overhead normally required for logical entangling gates. Yet this symmetry is highly restrictive: phantom codes are rare, the number of logical qubits is limited to $k=O(\\log n)$, and no phantom qLDPC family with growing logical dimension was previously known. We make three contributions. 1. We prove that recognizing phantomness of a given stabilizer code is at least as hard as Graph Isomorphism, even for CSS codes encoding only $k=2$ logical qubits. 2. We propose the first rate optimal phantom qLDPC families: for every fixed $D$, our CSS families achieve the maximal logical scaling $k=Θ(\\log n)$ and distance $d\\geq D$. 3. We prove a distance no-go theorem: every phantom code family with $k=ω(\\sqrt{\\log n})$ and check weight $w=O(1)$ satisfies $d\\leq w$ for sufficiently large $n$. Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling $k=Θ(\\log n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16542",
  "author_total": 3,
  "authors": [
    {
      "name": "Rui Mao"
    },
    {
      "name": "Weixiao Sun"
    },
    {
      "name": "Shengyu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9rj6yjthd223p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16542",
  "fulltext_url": null,
  "id": "9rj6yjthd223p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16542",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7213412706",
  "pdf_url": "https://arxiv.org/pdf/2609.16542",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9rj6yjthd223p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:41Z",
      "native_id": "2609.16542",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16542"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Phantom Codes: Hardness, Rate Optimal qLDPC Constructions, and Distance Limits",
  "title_text": "Phantom Codes: Hardness, Rate Optimal qLDPC Constructions, and Distance Limits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:41Z",
  "url": "https://research.draconitic.com/paper/9rj6yjthd223p",
  "venue": "arXiv",
  "versions": []
}