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  "slug": "ultrametric-passive-fault-tolerance-on-bruhat-tits-trees-analytic-deriv-93ee94f3",
  "title": "Ultrametric Passive Fault Tolerance on Bruhat–Tits Trees: Analytic Derivation and Quantitative Projections",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Quantum computing today is limited by the large physical‑to‑logical qubit overhead required for active error correction, which demands frequent syndrome extraction, low‑latency decoding, and extensive ancilla resources. A proposed alternative encodes logical information on the vertices of a p‑adic Bruhat–Tits tree, exploiting the ultrametric strong triangle inequality to confine error propagation geometrically. Existing literature lacks a rigorous analytic foundation for such “passive” protection and provides no quantitative comparison with surface‑code thresholds. We develop a self‑contained analytic model of an ultrametric ball code, derive exact combinatorial expressions for the required number of physical qudits, the logical distance, and the passive logical failure probability under independent depolarising noise. Using the model we compute concrete numbers for p=3 and tree depths t=1,2, obtaining a ball size n₁=5 qudits, passive logical failure P_pass=5.992003×10^-6, and a suppression factor S≈166.9 relative to an unencoded qudit. The results indicate that, at a physical error rate of 10^-3, the ultrametric scheme achieves logical error rates comparable to a distance‑3 surface code while using five‑fold fewer physical qudits. If the analytic assumptions hold, the approach would dramatically reduce qubit overhead and cooling requirements for fault‑tolerant quantum processors.",
  "abstract_markdown": "Quantum computing today is limited by the large physical‑to‑logical qubit overhead required for active error correction, which demands frequent syndrome extraction, low‑latency decoding, and extensive ancilla resources. A proposed alternative encodes logical information on the vertices of a $p$‑adic Bruhat–Tits tree, exploiting the ultrametric strong triangle inequality to confine error propagation geometrically. Existing literature lacks a rigorous analytic foundation for such “passive” protection and provides no quantitative comparison with surface‑code thresholds. We develop a self‑contained analytic model of an ultrametric ball code, derive exact combinatorial expressions for the required number of physical qudits, the logical distance, and the passive logical failure probability under independent depolarising noise. Using the model we compute concrete numbers for $p=3$ and tree depths $t=1,2$, obtaining a ball size $n_{1}=5$ qudits, passive logical failure $P_{\\mathrm{pass}}=5.992003\\times10^{-6}$, and a suppression factor $S\\approx166.9$ relative to an unencoded qudit. The results indicate that, at a physical error rate of $10^{-3}$, the ultrametric scheme achieves logical error rates comparable to a distance‑3 surface code while using five‑fold fewer physical qudits. If the analytic assumptions hold, the approach would dramatically reduce qubit overhead and cooling requirements for fault‑tolerant quantum processors.",
  "date": "2026-10-21",
  "first_published": "2026-10-21",
  "last_revised": "2026-10-21T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/ultrametric-passive-fault-tolerance-on-bruhat-tits-trees-analytic-deriv-93ee94f3",
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    "titles": [
      {
        "title": "Ultrametric Passive Fault Tolerance on Bruhat–Tits Trees: Analytic Derivation and Quantitative Projections"
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