{
  "id": "oai:papers.qnfo.org:passive-fault-tolerance-via-bruhat-tits-tree-codes-exact-ball-code-error-rates-e",
  "slug": "passive-fault-tolerance-via-bruhat-tits-tree-codes-exact-ball-code-error-rates-e",
  "title": "Passive Fault Tolerance via Bruhat–Tits Tree Codes: Exact Ball-Code Error Rates, Energy Barriers, and the Cooling Wall of Active Quantum Error Correction",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Large-scale quantum computation is gated by active error correction: stabilizer codes such as the surface code require physical-to-logical overheads of 10²–10⁴, continuous syndrome measurement, and millikelvin refrigeration whose thermodynamic cost grows as the operating temperature falls. A proposed alternative encodes logical qudits on balls of the Bruhat–Tits tree T_p+1, so that the ultrametric strong triangle inequality confines errors to single branches and correction is passive. The gap addressed here is quantitative: prior statements of the scheme assert a depolarizing threshold near 75\\% and removal of the cooling wall, but neither figure has been derived. We formalize the ball-code model, derive the exact sizes n_t = 1 + (p+1)(p^t-1)/(p-1), and compute with full arithmetic the passive logical failure probabilities P_pass(1) = 9.980×10^-6 per epoch for the p=3, t=1 block (5 qudits, distance 3) and P_pass(2) = 6.729×10^-7 for the t=2 block (17 qudits, distance 5) at ε = 10^-3, suppression factors of $100.2$ and $1486$. Under the conjectured barrier E_barrier(d)=2^d, thermal activation gives p_L ≤ 3.355×10^-4 at depth 3. The Carnot coefficient-of-performance ratio between 4 K and 10 mK operation is $405$. The 75\\% threshold remains an open conjecture; encoder existence is the binding constraint.",
  "abstract_markdown": "Large-scale quantum computation is gated by active error correction: stabilizer codes such as the surface code require physical-to-logical overheads of $10^{2}$–$10^{4}$, continuous syndrome measurement, and millikelvin refrigeration whose thermodynamic cost grows as the operating temperature falls. A proposed alternative encodes logical qudits on balls of the Bruhat–Tits tree $T_{p+1}$, so that the ultrametric strong triangle inequality confines errors to single branches and correction is passive. The gap addressed here is quantitative: prior statements of the scheme assert a depolarizing threshold near $75\\%$ and removal of the cooling wall, but neither figure has been derived. We formalize the ball-code model, derive the exact sizes $n_t = 1 + (p+1)(p^{t}-1)/(p-1)$, and compute with full arithmetic the passive logical failure probabilities $P_{\\mathrm{pass}}(1) = 9.980\\times10^{-6}$ per epoch for the $p=3$, $t=1$ block (5 qudits, distance 3) and $P_{\\mathrm{pass}}(2) = 6.729\\times10^{-7}$ for the $t=2$ block (17 qudits, distance 5) at $\\varepsilon = 10^{-3}$, suppression factors of $100.2$ and $1486$. Under the conjectured barrier $E_{\\text{barrier}}(d)=2^{d}$, thermal activation gives $p_L \\le 3.355\\times10^{-4}$ at depth 3. The Carnot coefficient-of-performance ratio between 4 K and 10 mK operation is $405$. The $75\\%$ threshold remains an open conjecture; encoder existence is the binding constraint.",
  "date": "2026-10-26",
  "first_published": "2026-10-26",
  "last_revised": "2026-10-26T00:00:00Z",
  "modified": "2026-10-26T00:00:00Z",
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  "quality_label": "High-quality scholarly research",
  "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/passive-fault-tolerance-via-bruhat-tits-tree-codes-exact-ball-code-error-rates-e",
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    "creators": [
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Passive Fault Tolerance via Bruhat–Tits Tree Codes: Exact Ball-Code Error Rates, Energy Barriers, and the Cooling Wall of Active Quantum Error Correction"
      }
    ],
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    "publicationYear": "2026",
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        "description": "Large-scale quantum computation is gated by active error correction: stabilizer codes such as the surface code require physical-to-logical overheads of 10²–10⁴, continuous syndrome measurement, and millikelvin refrigeration whose thermodynamic cost grows as the operating temperature falls. A proposed alternative encodes logical qudits on balls of the Bruhat–Tits tree T_p+1, so that the ultrametric strong triangle inequality confines errors to single branches and correction is passive. The gap addressed here is quantitative: prior statements of the scheme assert a depolarizing threshold near 75\\% and removal of the cooling wall, but neither figure has been derived. We formalize the ball-code model, derive the exact sizes n_t = 1 + (p+1)(p^t-1)/(p-1), and compute with full arithmetic the passive logical failure probabilities P_pass(1) = 9.980×10^-6 per epoch for the p=3, t=1 block (5 qudits, distance 3) and P_pass(2) = 6.729×10^-7 for the t=2 block (17 qudits, distance 5) at ε = 10^-3, suppression factors of $100.2$ and $1486$. Under the conjectured barrier E_barrier(d)=2^d, thermal activation gives p_L ≤ 3.355×10^-4 at depth 3. The Carnot coefficient-of-performance ratio between 4 K and 10 mK operation is $405$. The 75\\% threshold remains an open conjecture; encoder existence is the binding constraint.",
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