{
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  "slug": "holographic-quantum-error-correction-on-bruhat-tits-trees-a-conditional-threshol",
  "title": "Holographic Quantum Error Correction on Bruhat–Tits Trees: A Conditional Threshold and Resource Analysis at 10^-4",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The holographic quantum error‑correction (QEC) architecture that implements encoding as a renormalization‑group (RG) flow on a Bruhat–Tits tree T_p claims a physical error‑rate threshold ε_th≥10^-4. No hardware data accompany the proposal, so we construct a fully explicit conditional model: (i) independent site errors on a regular tree of valency q=p+1, (ii) a concatenation‑style logical‑error law ε_L=A(ε/ε_th)^(d+1)/2 with A=1, (iii) a reliability target P_fail≤0.01 over T=10⁹ correction cycles at a cycle rate f_c=10⁶ mathrms^-1. Solving the model yields (a) zero logical protection at ε=ε_th=10^-4, (b) a distance d≥21 code sufficient at ε=10^-5, requiring 88,573 physical sites per logical qubit and a measurement bandwidth of 8.8573×10¹0 mathrms^-1, (c) a shallow d=11 code viable only if ε≤1.4678×10^-6, and (d) an exponential resource explosion when operating at ε=5×10^-5. An alternative quadratic‑map convention shows the threshold equals 1/κ, so the 10^-4 target holds iff κ≤10⁴. The analysis demonstrates internal consistency of the threshold claim but reveals extreme resource demands; the claim’s fate hinges on measurable quantities κ, A, and error correlations that the original proposal does not provide.",
  "abstract_markdown": "The holographic quantum error‑correction (QEC) architecture that implements encoding as a renormalization‑group (RG) flow on a Bruhat–Tits tree $\\mathcal{T}_p$ claims a physical error‑rate threshold $\\epsilon_{th}\\ge10^{-4}$. No hardware data accompany the proposal, so we construct a fully explicit conditional model: (i) independent site errors on a regular tree of valency $q=p+1$, (ii) a concatenation‑style logical‑error law $\\epsilon_L=A(\\epsilon/\\epsilon_{th})^{(d+1)/2}$ with $A=1$, (iii) a reliability target $P_{fail}\\le0.01$ over $T=10^{9}$ correction cycles at a cycle rate $f_c=10^{6}\\,\\mathrm{s^{-1}}$. Solving the model yields (a) zero logical protection at $\\epsilon=\\epsilon_{th}=10^{-4}$, (b) a distance $d\\ge21$ code sufficient at $\\epsilon=10^{-5}$, requiring $88{,}573$ physical sites per logical qubit and a measurement bandwidth of $8.8573\\times10^{10}\\,\\mathrm{s^{-1}}$, (c) a shallow $d=11$ code viable only if $\\epsilon\\le1.4678\\times10^{-6}$, and (d) an exponential resource explosion when operating at $\\epsilon=5\\times10^{-5}$. An alternative quadratic‑map convention shows the threshold equals $1/\\kappa$, so the $10^{-4}$ target holds iff $\\kappa\\le10^{4}$. The analysis demonstrates internal consistency of the threshold claim but reveals extreme resource demands; the claim’s fate hinges on measurable quantities $\\kappa$, $A$, and error correlations that the original proposal does not provide.",
  "date": "2026-10-12",
  "first_published": "2026-10-12",
  "last_revised": "2026-10-19T12:21:04Z",
  "modified": "2026-10-19T12:21:04Z",
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  "url": "https://papers.qnfo.org/papers/holographic-quantum-error-correction-on-bruhat-tits-trees-a-conditional-threshol",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Holographic Quantum Error Correction on Bruhat–Tits Trees: A Conditional Threshold and Resource Analysis at 10^-4"
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        "date": "2026-10-12",
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        "description": "The holographic quantum error‑correction (QEC) architecture that implements encoding as a renormalization‑group (RG) flow on a Bruhat–Tits tree T_p claims a physical error‑rate threshold ε_th≥10^-4. No hardware data accompany the proposal, so we construct a fully explicit conditional model: (i) independent site errors on a regular tree of valency q=p+1, (ii) a concatenation‑style logical‑error law ε_L=A(ε/ε_th)^(d+1)/2 with A=1, (iii) a reliability target P_fail≤0.01 over T=10⁹ correction cycles at a cycle rate f_c=10⁶ mathrms^-1. Solving the model yields (a) zero logical protection at ε=ε_th=10^-4, (b) a distance d≥21 code sufficient at ε=10^-5, requiring 88,573 physical sites per logical qubit and a measurement bandwidth of 8.8573×10¹0 mathrms^-1, (c) a shallow d=11 code viable only if ε≤1.4678×10^-6, and (d) an exponential resource explosion when operating at ε=5×10^-5. An alternative quadratic‑map convention shows the threshold equals 1/κ, so the 10^-4 target holds iff κ≤10⁴. The analysis demonstrates internal consistency of the threshold claim but reveals extreme resource demands; the claim’s fate hinges on measurable quantities κ, A, and error correlations that the original proposal does not provide.",
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