{
  "abstract": "We study fault-tolerant quantum computation using a fixed, local error-correction rule and a geometrically simple code: the cubical $(d,d')$-Toric code with $d',d-d'\\geq 2$, including the 4D Toric code. Under sufficiently weak local-stochastic Pauli noise, our construction uses noiseless, bounded-depth classical processing per physical time step. Its correction cycle acts only on the current syndrome, has constant depth, and is repeated unchanged during idle logical computation; it requires neither multiscale decoder states nor scale-dependent communication delays. Our proof treats errors and syndromes as chains in a cubical complex, combining geometric shrinking under the sweep rule with a space--time explanation-tree argument inspired by Toom's theorem. It does not rely on expansion. We establish three results: 1.Explicit higher-dimensional sweep rule. We give a local algorithm for choosing sweep corrections on cubical $(d,d')$-Toric complexes, resolving the local-existence question of Kubica and Preskill. 2. Fault-tolerant computation. We prove a circuit-level threshold theorem for noisy sweep cycles interleaved with logical gates. In particular, the 4D Toric code admits a fixed, constant-depth syndrome-based correction cycle without a multiscale decoder or expansion-based encoding. We do not claim constant total-depth overhead. 3. Ideal decoding under local-stochastic noise. For an initial local-stochastic Pauli error of rate $p$ below threshold, $n^{d+1}$ noiseless sweep cycles recover the encoded state on a side-length-$n$ $(d,d')$-Toric code with probability at least $1-O(\\operatorname{poly}(n))p^{Ω(n)}$, extending earlier guarantees for independent errors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38847",
  "author_total": 1,
  "authors": [
    {
      "name": "David Ponarovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/brqd0t8g30ccy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38847",
  "fulltext_url": null,
  "id": "brqd0t8g30ccy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38847",
  "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.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/brqd0t8g30ccy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.38847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38847"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Local Quantum Fault Tolerance Without Expansion",
  "title_text": "Local Quantum Fault Tolerance Without Expansion",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:17Z",
  "url": "https://research.draconitic.com/paper/brqd0t8g30ccy",
  "venue": "arXiv",
  "versions": []
}