{
  "abstract": "We construct an explicit transition matrix that exactly describes code-adapted basis-population dynamics of stabilizer codes under circuit-level stochastic Pauli noise. The matrix is expressed in a code-adapted basis that captures probability flow between logical-basis labels and syndrome sectors. For quantum error detection (QED), we incorporate post-selection by aggregating rejected outcomes into an absorbing state, so that a single transition matrix represents a full clock cycle of gates, syndrome extraction, and post-selection. Additionally, we characterize when protocol symmetries permit exact lumping to simpler, more interpretable models. The transition matrix framework enables direct application of classical stochastic-matrix techniques to analyze multi-cycle QED protocols. First, we identify a leading-order obstruction to a stationary logical-only accepted-map description, and quantify the accepted leakage injection from QED-check imperfections that causes it. Second, we express leading-order QED efficiency as a function of check frequency in terms of physically meaningful parameters, and prove that less frequent checks improve efficiency to leading order. We evaluate the framework on the $[[4,2,2]]$ and $[[5,1,3]]$ codes under depolarizing gate noise and readout errors: transition-matrix predictions agree with million-shot Stim Monte Carlo at the scale of the reported confidence intervals, and matrix-computed results illustrate the first-cycle transient and interval-dependent efficiency. More broadly, the transition-matrix formalism provides both an analytical foundation for QED analysis and an interpretable tool for understanding code behavior under realistic noise.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12073",
  "author_total": 3,
  "authors": [
    {
      "name": "Rohan S. Kumar"
    },
    {
      "name": "Ben Foxman"
    },
    {
      "name": "Yongshan Ding"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ed6g6j86a1kw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12073",
  "fulltext_url": null,
  "id": "0ed6g6j86a1kw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12073",
  "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": "W7213016725",
  "pdf_url": "https://arxiv.org/pdf/2609.12073",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ed6g6j86a1kw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:07Z",
      "native_id": "2609.12073",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12073"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Modeling Quantum Error Detection with Transition Matrices",
  "title_text": "Modeling Quantum Error Detection with Transition Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:07Z",
  "url": "https://research.draconitic.com/paper/0ed6g6j86a1kw",
  "venue": "arXiv",
  "versions": []
}