{
  "id": "oai:papers.qnfo.org:ultrametric-passive-fault-tolerance-on-bruhat-tits-trees-analytic-derivation-and",
  "slug": "ultrametric-passive-fault-tolerance-on-bruhat-tits-trees-analytic-derivation-and",
  "title": "Ultrametric Passive Fault Tolerance on Bruhat–Tits Trees: Analytic Derivation and Numerical Projections",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Fault‑tolerant quantum computation currently demands a physical‑to‑logical qubit overhead of 10²–10⁴, a cost that limits near‑term scalability. A major gap is the lack of a passive protection mechanism that does not require syndrome extraction or additional ancilla qubits. We analytically encode logical information on a p‑adic Bruhat–Tits tree, exploiting the strong triangle inequality of ultrametric spaces to confine error propagation geometrically. By deriving the energy‑barrier scaling E_barrier(d)=2^d for tree depth d, establishing exact code‑parameter formulas n_t=1+frac(p+1)(p^t-1)p-1, and evaluating independent depolarizing noise with rate ε, we obtain a passive logical failure probability P_pass≈9.97×10^-6 per protection epoch for p=3, t=1, ε=10^-3, corresponding to a suppression factor of $100.3$ over an unencoded qudit. A projection that combines the ultrametric code with q≈128 atomic hyperfine levels yields an additional 48× reduction, giving a logical error estimate ∼2.1×10^-7. These results suggest that ultrametric geometry can reduce qubit overhead by two orders of magnitude without active error correction, assuming a physical‑to‑logical qubit overhead of 10²–10⁴, potentially reshaping hardware design and cloud‑service economics.",
  "abstract_markdown": "Fault‑tolerant quantum computation currently demands a physical‑to‑logical qubit overhead of $10^{2}$–$10^{4}$, a cost that limits near‑term scalability. A major gap is the lack of a passive protection mechanism that does not require syndrome extraction or additional ancilla qubits. We analytically encode logical information on a $p$‑adic Bruhat–Tits tree, exploiting the strong triangle inequality of ultrametric spaces to confine error propagation geometrically. By deriving the energy‑barrier scaling $E_{\\text{barrier}}(d)=2^{d}$ for tree depth $d$, establishing exact code‑parameter formulas $n_{t}=1+\\frac{(p+1)(p^{t}-1)}{p-1}$, and evaluating independent depolarizing noise with rate $\\varepsilon$, we obtain a passive logical failure probability $P_{\\mathrm{pass}}\\approx9.97\\times10^{-6}$ per protection epoch for $p=3$, $t=1$, $\\varepsilon=10^{-3}$, corresponding to a suppression factor of $100.3$ over an unencoded qudit. A projection that combines the ultrametric code with $q\\approx128$ atomic hyperfine levels yields an additional $48\\times$ reduction, giving a logical error estimate $\\sim2.1\\times10^{-7}$. These results suggest that ultrametric geometry can reduce qubit overhead by two orders of magnitude without active error correction, assuming a physical‑to‑logical qubit overhead of $10^{2}$–$10^{4}$, potentially reshaping hardware design and cloud‑service economics.",
  "date": "2026-10-20",
  "first_published": "2026-10-20",
  "last_revised": "2026-10-20T00:00:00Z",
  "modified": "2026-10-20T00:00:00Z",
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  "doi": null,
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  "url": "https://papers.qnfo.org/papers/ultrametric-passive-fault-tolerance-on-bruhat-tits-trees-analytic-derivation-and",
  "license_url": "https://legal.qnfo.org/",
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    "creators": [
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Ultrametric Passive Fault Tolerance on Bruhat–Tits Trees: Analytic Derivation and Numerical Projections"
      }
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    "publicationYear": "2026",
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        "description": "Fault‑tolerant quantum computation currently demands a physical‑to‑logical qubit overhead of 10²–10⁴, a cost that limits near‑term scalability. A major gap is the lack of a passive protection mechanism that does not require syndrome extraction or additional ancilla qubits. We analytically encode logical information on a p‑adic Bruhat–Tits tree, exploiting the strong triangle inequality of ultrametric spaces to confine error propagation geometrically. By deriving the energy‑barrier scaling E_barrier(d)=2^d for tree depth d, establishing exact code‑parameter formulas n_t=1+frac(p+1)(p^t-1)p-1, and evaluating independent depolarizing noise with rate ε, we obtain a passive logical failure probability P_pass≈9.97×10^-6 per protection epoch for p=3, t=1, ε=10^-3, corresponding to a suppression factor of $100.3$ over an unencoded qudit. A projection that combines the ultrametric code with q≈128 atomic hyperfine levels yields an additional 48× reduction, giving a logical error estimate ∼2.1×10^-7. These results suggest that ultrametric geometry can reduce qubit overhead by two orders of magnitude without active error correction, assuming a physical‑to‑logical qubit overhead of 10²–10⁴, potentially reshaping hardware design and cloud‑service economics.",
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