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  "id": "oai:papers.qnfo.org:adelic-quantum-error-correction-a-synthesis-across-all-primes-v1-0-0",
  "slug": "adelic-quantum-error-correction-a-synthesis-across-all-primes-v1-0-0",
  "title": "Adelic Quantum Error Correction: A Synthesis Across All Primes v1.0.0",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The p-adic quantum error correction program has developed three independent but connected strands: a mathematical theory of p-adic stabilizer codes (Phase A), an experimental protocol to test for ultrametric error structure (Phase B), and a hardware co-design proposal for tree-structured qubit layouts (Phase C). These strands share a common origin in the ultrametric geometry of the p-adic numbers Q_p, but they treat each prime p in isolation. Number theory provides a deeper unification: the adele ring A, which stitches together all completions of Q — the real numbers R (Archimedean) and all p-adic fields Q_p (non-Archimedean) — into a single locally compact topological ring. The adelic product formula ∏_p ≤ ∞ |x|_p = 1 implies that contributions from all primes are balanced: what is gained in one completion is lost in another. We propose that quantum error correction should be formulated on the adele ring rather than on any single completion. In this framework, a physical qubit lives in real (Archimedean) spacetime but its errors exhibit ultrametric structure across a hierarchy of primes. [...]",
  "abstract_markdown": "The $p$-adic quantum error correction program has developed three independent but connected strands: a mathematical theory of $p$-adic stabilizer codes (Phase A), an experimental protocol to test for ultrametric error structure (Phase B), and a hardware co-design proposal for tree-structured qubit layouts (Phase C). These strands share a common origin in the ultrametric geometry of the $p$-adic numbers $\\mathbb{Q}_p$, but they treat each prime $p$ in isolation. Number theory provides a deeper unification: the adele ring $\\mathbb{A}$, which stitches together all completions of $\\mathbb{Q}$ — the real numbers $\\mathbb{R}$ (Archimedean) and all $p$-adic fields $\\mathbb{Q}_p$ (non-Archimedean) — into a single locally compact topological ring. The adelic product formula $\\prod_{p \\leq \\infty} |x|_p = 1$ implies that contributions from all primes are balanced: what is gained in one completion is lost in another. We propose that quantum error correction should be formulated on the adele ring rather than on any single completion. In this framework, a physical qubit lives in real (Archimedean) spacetime but its errors exhibit ultrametric structure across a hierarchy of primes. [...]",
  "date": "2026-07-27",
  "first_published": "2026-07-27",
  "last_revised": "2026-07-27T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/adelic-quantum-error-correction-a-synthesis-across-all-primes-v1-0-0",
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    "titles": [
      {
        "title": "Adelic Quantum Error Correction: A Synthesis Across All Primes v1.0.0"
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    "publicationYear": "2026",
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    "dates": [
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        "description": "The p-adic quantum error correction program has developed three independent but connected strands: a mathematical theory of p-adic stabilizer codes (Phase A), an experimental protocol to test for ultrametric error structure (Phase B), and a hardware co-design proposal for tree-structured qubit layouts (Phase C). These strands share a common origin in the ultrametric geometry of the p-adic numbers Q_p, but they treat each prime p in isolation. Number theory provides a deeper unification: the adele ring A, which stitches together all completions of Q — the real numbers R (Archimedean) and all p-adic fields Q_p (non-Archimedean) — into a single locally compact topological ring. The adelic product formula ∏_p ≤ ∞ |x|_p = 1 implies that contributions from all primes are balanced: what is gained in one completion is lost in another. We propose that quantum error correction should be formulated on the adele ring rather than on any single completion. In this framework, a physical qubit lives in real (Archimedean) spacetime but its errors exhibit ultrametric structure across a hierarchy of primes. [...]",
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