{
  "id": "oai:papers.qnfo.org:passive-quantum-error-correction-via-ultrametric-geometry-on-bruhat-tits-trees",
  "slug": "passive-quantum-error-correction-via-ultrametric-geometry-on-bruhat-tits-trees",
  "title": "Passive Quantum Error Correction via Ultrametric Geometry on Bruhat–Tits Trees",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Scalable quantum computing is limited by the thermodynamic overhead of active error‑correction cycles, which require repeated syndrome extraction, low‑latency decoding, and large ancilla resources. A quantitative gap exists: no fully derived account shows how non‑Archimedean (ultrametric) geometry can passively confine errors, compute exact code parameters, logical‑failure probabilities, and the associated cooling budget. We formalize logical qudits on finite balls of the Bruhat–Tits tree T_p+1, derive the exact ball‑size formula n_t=1+frac(p+1)(p^t-1)p-1, the conditional distance d=2t+1, and the barrier height B(d)=lceil p/2rceil^d (which equals 2^d for p=3) for odd p. Analytic evaluation of the passive logical‑failure probability yields P_pass=9.98×10^-6 for p=3, t=1, ε=10^-3 and P_pass^(2)=6.73×10^-7 for t=2. Barrier‑based logical‑error bounds p_L(3)=6.5536×10^-4 and p_L(4)=4.295×10^-7 follow from p_L(d)=p_err^ B(d) with p_err=0.40. Thermal‑activation analysis gives p_L(d)≤ e^-B(d) at k_BT=1, and a cooling‑budget requirement Δ≥1.727 k_BT per site for a 10^-12 logical‑error target at depth 4. These results convert the ultrametric passive‑QEC conjecture into explicit, falsifiable numbers and identify symmetric‑encoder construction as the remaining open task.",
  "abstract_markdown": "Scalable quantum computing is limited by the thermodynamic overhead of active error‑correction cycles, which require repeated syndrome extraction, low‑latency decoding, and large ancilla resources. A quantitative gap exists: no fully derived account shows how non‑Archimedean (ultrametric) geometry can passively confine errors, compute exact code parameters, logical‑failure probabilities, and the associated cooling budget. We formalize logical qudits on finite balls of the Bruhat–Tits tree $T_{p+1}$, derive the exact ball‑size formula $n_t=1+\\frac{(p+1)(p^{t}-1)}{p-1}$, the conditional distance $d=2t+1$, and the barrier height $B(d)=\\lceil p/2\\rceil^{d}$ (which equals $2^{d}$ for $p=3$) for odd $p$. Analytic evaluation of the passive logical‑failure probability yields $P_{\\mathrm{pass}}=9.98\\times10^{-6}$ for $p=3$, $t=1$, $\\varepsilon=10^{-3}$ and $P_{\\mathrm{pass}}^{(2)}=6.73\\times10^{-7}$ for $t=2$. Barrier‑based logical‑error bounds $p_L(3)=6.5536\\times10^{-4}$ and $p_L(4)=4.295\\times10^{-7}$ follow from $p_L(d)=p_{\\mathrm{err}}^{\\,B(d)}$ with $p_{\\mathrm{err}}=0.40$. Thermal‑activation analysis gives $p_L(d)\\le e^{-B(d)}$ at $k_{\\mathrm{B}}T=1$, and a cooling‑budget requirement $\\Delta\\ge1.727\\,k_{\\mathrm{B}}T$ per site for a $10^{-12}$ logical‑error target at depth 4. These results convert the ultrametric passive‑QEC conjecture into explicit, falsifiable numbers and identify symmetric‑encoder construction as the remaining open task.",
  "date": "2026-10-21",
  "first_published": "2026-10-21",
  "last_revised": "2026-10-21T00:00:00Z",
  "modified": "2026-10-21T00:00:00Z",
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  "language": "en",
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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/passive-quantum-error-correction-via-ultrametric-geometry-on-bruhat-tits-trees",
  "license_url": "https://legal.qnfo.org/",
  "license_spdx": null,
  "sha256": "2967a973527686ef79b270c7453bed66f84e58d1c399e6764d4923400196c7bd",
  "version_url": "https://papers.qnfo.org/papers/passive-quantum-error-correction-via-ultrametric-geometry-on-bruhat-tits-trees/v/1.0.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Passive Quantum Error Correction via Ultrametric Geometry on Bruhat–Tits Trees"
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    "publicationYear": "2026",
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        "date": "2026-10-21",
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        "description": "Scalable quantum computing is limited by the thermodynamic overhead of active error‑correction cycles, which require repeated syndrome extraction, low‑latency decoding, and large ancilla resources. A quantitative gap exists: no fully derived account shows how non‑Archimedean (ultrametric) geometry can passively confine errors, compute exact code parameters, logical‑failure probabilities, and the associated cooling budget. We formalize logical qudits on finite balls of the Bruhat–Tits tree T_p+1, derive the exact ball‑size formula n_t=1+frac(p+1)(p^t-1)p-1, the conditional distance d=2t+1, and the barrier height B(d)=lceil p/2rceil^d (which equals 2^d for p=3) for odd p. Analytic evaluation of the passive logical‑failure probability yields P_pass=9.98×10^-6 for p=3, t=1, ε=10^-3 and P_pass^(2)=6.73×10^-7 for t=2. Barrier‑based logical‑error bounds p_L(3)=6.5536×10^-4 and p_L(4)=4.295×10^-7 follow from p_L(d)=p_err^ B(d) with p_err=0.40. Thermal‑activation analysis gives p_L(d)≤ e^-B(d) at k_BT=1, and a cooling‑budget requirement Δ≥1.727 k_BT per site for a 10^-12 logical‑error target at depth 4. These results convert the ultrametric passive‑QEC conjecture into explicit, falsifiable numbers and identify symmetric‑encoder construction as the remaining open task.",
        "descriptionType": "Abstract"
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