{
  "abstract": "We construct quantum LDPC (qLDPC) and high-rate CSS codes from our recently introduced classical fair-density parity-check (FDPC) codes. To this end, we introduce a structured sparsification of FDPC parity-check matrices, which reduces their check weights while preserving the underlying combinatorial structure and distance guarantees. Combined with the hypergraph-product construction, this yields finite-length qLDPC codes with analytically controlled blocklength, dimension, certified distance, and stabilizer weight. For quantum blocklengths $N<10^5$, the constructions introduced here span guaranteed rates from approximately $0.35\\%$ to $25.8\\%$ and certified quantum distances from $12$ to $69$, with stabilizer weights between $8$ and $16$. In the large-blocklength regime, allowing the FDPC order and sparsified check weight to scale moderately with blocklength yields a family of high-rate CSS codes, which we refer to as quantum FDPC (qFDPC) codes, with rate $R_Q$ and minimum distance $D$ satisfying \\[ R_Q = 1-O\\left(\\frac{1}{\\log\\log N}\\right), \\ \\ D=Ω(N^{1/4}), \\] and stabilizer weight $O(\\log N\\log\\log N)$. Finally, the analytically available FDPC weight distribution provides explicit information about the logical operators of the resulting hypergraph-product codes. Over the quantum erasure channel, this structure yields rigorous first-order maximum likelihood (ML) expressions and a higher-order weight-distribution approximation to the ML logical block error probability. This enables estimation of finite-length operating points and the onset of the error-floor regime. To the best of our knowledge, beyond surface-code-type constructions, this is the first finite-rate qLDPC framework to provide both finite-length certified minimum-distance information and an analytical characterization of low-weight logical multiplicities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01181",
  "author_total": 1,
  "authors": [
    {
      "name": "Hessam Mahdavifar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bzgz84x3m44x5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01181",
  "fulltext_url": null,
  "id": "bzgz84x3m44x5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01181",
  "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"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7206152168",
  "pdf_url": "https://arxiv.org/pdf/2609.01181",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bzgz84x3m44x5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01181",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01181"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Quantum LDPC and High-Rate CSS Codes from Fair-Density Parity-Check Codes",
  "title_text": "Quantum LDPC and High-Rate CSS Codes from Fair-Density Parity-Check Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/bzgz84x3m44x5",
  "venue": "arXiv",
  "versions": []
}