{
  "id": "oai:papers.qnfo.org:ultrametric-passive-error-correction-on-p-ary-bruhat-tits-trees-barrier-analysis",
  "slug": "ultrametric-passive-error-correction-on-p-ary-bruhat-tits-trees-barrier-analysis",
  "title": "Ultrametric Passive Error Correction on p-ary Bruhat–Tits Trees: Barrier Analysis and Recursive Thresholds",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Fault‑tolerant quantum computation presently depends on active stabilizer codes such as the surface code, which require repeated syndrome extraction, low‑latency decoding, and large ancilla overhead. A fundamentally different approach would encode logical qubits on the vertices of a regular odd‑p Bruhat–Tits tree and let the ultrametric geometry enforce a passive error barrier. The gap is the absence of a quantitative analysis that connects the tree depth d to an explicit logical‑error suppression factor. We therefore (i) derive the barrier height B(d)=lceil p/2rceil^d from the strong triangle inequality of the underlying ultrametric, (ii) propagate this barrier into a logical‑error bound p_L(d)≤ C p_err^B(d) with concrete constants, and (iii) evaluate the bound for the physically relevant case p=3, p_err=0.40 and depths d=3,4. The principal results are: (a) B(3)=2³=8 and p_L(3)≤6.55×10^-4; (b) B(4)=2⁴=16 and p_L(4)≤4.29×10^-7; (c) a recursive eight‑fold majority‑vote decoding scheme that reduces the physical error probability from 0.40 to 1.7367×10^-1, then to 5.55×10^-3, and finally to 2.91×10^-10 over three concatenation levels (matching the numbers reported in [8]). If the barrier model holds, quantum hardware designers could replace costly syndrome‑extraction circuitry with static connectivity and majority‑vote decoding, dramatically lowering control‑system complexity.",
  "abstract_markdown": "Fault‑tolerant quantum computation presently depends on active stabilizer codes such as the surface code, which require repeated syndrome extraction, low‑latency decoding, and large ancilla overhead. A fundamentally different approach would encode logical qubits on the vertices of a regular odd‑\\(p\\) Bruhat–Tits tree and let the ultrametric geometry enforce a passive error barrier. The gap is the absence of a quantitative analysis that connects the tree depth \\(d\\) to an explicit logical‑error suppression factor. We therefore (i) derive the barrier height \\(B(d)=\\lceil p/2\\rceil^{d}\\) from the strong triangle inequality of the underlying ultrametric, (ii) propagate this barrier into a logical‑error bound \\(p_{L}(d)\\le C\\,p_{\\mathrm{err}}^{B(d)}\\) with concrete constants, and (iii) evaluate the bound for the physically relevant case \\(p=3\\), \\(p_{\\mathrm{err}}=0.40\\) and depths \\(d=3,4\\). The principal results are: (a) \\(B(3)=2^{3}=8\\) and \\(p_{L}(3)\\le6.55\\times10^{-4}\\); (b) \\(B(4)=2^{4}=16\\) and \\(p_{L}(4)\\le4.29\\times10^{-7}\\); (c) a recursive eight‑fold majority‑vote decoding scheme that reduces the physical error probability from \\(0.40\\) to \\(1.7367\\times10^{-1}\\), then to \\(5.55\\times10^{-3}\\), and finally to \\(2.91\\times10^{-10}\\) over three concatenation levels (matching the numbers reported in [8]). If the barrier model holds, quantum hardware designers could replace costly syndrome‑extraction circuitry with static connectivity and majority‑vote decoding, dramatically lowering control‑system complexity.",
  "date": "2026-10-18",
  "first_published": "2026-10-18",
  "last_revised": "2026-10-18T00:00:00Z",
  "modified": "2026-10-18T00:00:00Z",
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  "language": "en",
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    "High-quality scholarly research",
    "open-access scholarly work",
    "versioned research publication"
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  "quality_basis": "QNFO publisher classification for open-access, versioned research with a public revision history and reader correction channel; this is not a claim of independent peer review.",
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  "peer_review_status": "Not independently peer-reviewed",
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  "doi": null,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/ultrametric-passive-error-correction-on-p-ary-bruhat-tits-trees-barrier-analysis",
  "license_url": "https://legal.qnfo.org/",
  "license_spdx": null,
  "sha256": "71d0abaa6240648e849025edfce9c26687d982374cd4b83d6c91542312d33f4d",
  "version_url": "https://papers.qnfo.org/papers/ultrametric-passive-error-correction-on-p-ary-bruhat-tits-trees-barrier-analysis/v/1.0.1",
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        "identifier": "https://papers.qnfo.org/papers/ultrametric-passive-error-correction-on-p-ary-bruhat-tits-trees-barrier-analysis",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Ultrametric Passive Error Correction on p-ary Bruhat–Tits Trees: Barrier Analysis and Recursive Thresholds"
      }
    ],
    "publisher": "QNFO",
    "publicationYear": "2026",
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    "dates": [
      {
        "date": "2026-10-18",
        "dateType": "Issued"
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        "date": "2026-10-18",
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        "description": "Fault‑tolerant quantum computation presently depends on active stabilizer codes such as the surface code, which require repeated syndrome extraction, low‑latency decoding, and large ancilla overhead. A fundamentally different approach would encode logical qubits on the vertices of a regular odd‑p Bruhat–Tits tree and let the ultrametric geometry enforce a passive error barrier. The gap is the absence of a quantitative analysis that connects the tree depth d to an explicit logical‑error suppression factor. We therefore (i) derive the barrier height B(d)=lceil p/2rceil^d from the strong triangle inequality of the underlying ultrametric, (ii) propagate this barrier into a logical‑error bound p_L(d)≤ C p_err^B(d) with concrete constants, and (iii) evaluate the bound for the physically relevant case p=3, p_err=0.40 and depths d=3,4. The principal results are: (a) B(3)=2³=8 and p_L(3)≤6.55×10^-4; (b) B(4)=2⁴=16 and p_L(4)≤4.29×10^-7; (c) a recursive eight‑fold majority‑vote decoding scheme that reduces the physical error probability from 0.40 to 1.7367×10^-1, then to 5.55×10^-3, and finally to 2.91×10^-10 over three concatenation levels (matching the numbers reported in [8]). If the barrier model holds, quantum hardware designers could replace costly syndrome‑extraction circuitry with static connectivity and majority‑vote decoding, dramatically lowering control‑system complexity.",
        "descriptionType": "Abstract"
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        "description": "Publisher quality classification: High-quality scholarly research. QNFO publisher classification for open-access, versioned research with a public revision history and reader correction channel; this is not a claim of independent peer review.",
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