{
  "abstract": "Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of $\\lceil d/2\\rceil$ for distance-$d$ codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed $T$ gate on the $[[16,4,4]]$ BB code, $T$-type rotations on the $[[64,8,8]]$ BB code, and CCZ-, CS-, and $T$-type operations on the $[[36,6,6]]$ Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields C${}^r$Z gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17612",
  "author_total": 5,
  "authors": [
    {
      "name": "Hunter Nelson"
    },
    {
      "name": "Arshag Danageozian"
    },
    {
      "name": "ChunJun Cao"
    },
    {
      "name": "Sophia E. Economou"
    },
    {
      "name": "Edwin Barnes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jfr993qr47rp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17612",
  "fulltext_url": null,
  "id": "6jfr993qr47rp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17612",
  "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": "W7221812330",
  "pdf_url": "https://arxiv.org/pdf/2610.17612",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jfr993qr47rp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:11Z",
      "native_id": "2610.17612",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17612"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "From Syndromes to Computation: A General Framework for Logical Operations in Quantum Error-Correcting Codes",
  "title_text": "From Syndromes to Computation: A General Framework for Logical Operations in Quantum Error-Correcting Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:11Z",
  "url": "https://research.draconitic.com/paper/6jfr993qr47rp",
  "venue": "arXiv",
  "versions": []
}