{
  "id": "oai:papers.qnfo.org:ultrametric-p-adic-noise-structure-and-heisenberg-limited-quantum-metrology-erro",
  "slug": "ultrametric-p-adic-noise-structure-and-heisenberg-limited-quantum-metrology-erro",
  "title": "Ultrametric p‑Adic Noise Structure and Heisenberg‑Limited Quantum Metrology: Error Resilience by Geometry, Not Overhead",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Quantum metrology seeks phase sensitivities that surpass the standard quantum limit (SQL) Δθ∝ N^-1/2, aiming for the Heisenberg limit Δθ∝ N^-1. Decoherence typically destroys this advantage, and active error‑correction incurs prohibitive overhead for near‑term sensors. This work addresses the gap that no passive noise‑geometry mechanism has been quantitatively analyzed for metrological scaling. By modeling decoherence as a p‑adic hierarchical stochastic process on an ultrametric lattice, the strong triangle inequality forces error events on nested clusters to be nested rather than independent. For a hierarchy of L levels with level‑k error weight ε_k=ε₀p^-k, the ultrametric error probability is P_U=ε₀p^-L, while the Archimedean independent‑event union bound is P_A=ε₀frac1-p^-(L+1)1-p^-1. With p=2, L=10, ε₀=10^-2 the ultrametric probability is 9.765625×10^-6 versus an Archimedean bound of 1.9990234375×10^-2, a ratio of exactly $2047$. Propagating these probabilities to the quantum Fisher information (QFI) of an N‑probe GHZ state with L=log₂N yields Heisenberg scaling Δθ∝ N^-1 under the ultrametric model, whereas independent dephasing collapses GHZ coherence exponentially in N. The principal result is a scaling prediction that is falsifiable by measuring nested‑event statistics; if the prediction fails, the ultrametric hypothesis is rejected. The analysis connects the model to p‑adic mathematical‑physics literature and documents the conventions adopted.",
  "abstract_markdown": "Quantum metrology seeks phase sensitivities that surpass the standard quantum limit (SQL) $\\Delta\\theta\\propto N^{-1/2}$, aiming for the Heisenberg limit $\\Delta\\theta\\propto N^{-1}$. Decoherence typically destroys this advantage, and active error‑correction incurs prohibitive overhead for near‑term sensors. This work addresses the gap that no passive noise‑geometry mechanism has been quantitatively analyzed for metrological scaling. By modeling decoherence as a $p$‑adic hierarchical stochastic process on an ultrametric lattice, the strong triangle inequality forces error events on nested clusters to be nested rather than independent. For a hierarchy of $L$ levels with level‑$k$ error weight $\\varepsilon_k=\\varepsilon_0p^{-k}$, the ultrametric error probability is $P_{\\mathrm{U}}=\\varepsilon_0p^{-L}$, while the Archimedean independent‑event union bound is $P_{\\mathrm{A}}=\\varepsilon_0\\frac{1-p^{-(L+1)}}{1-p^{-1}}$. With $p=2$, $L=10$, $\\varepsilon_0=10^{-2}$ the ultrametric probability is $9.765625\\times10^{-6}$ versus an Archimedean bound of $1.9990234375\\times10^{-2}$, a ratio of exactly $2047$. Propagating these probabilities to the quantum Fisher information (QFI) of an $N$‑probe GHZ state with $L=\\log_2N$ yields Heisenberg scaling $\\Delta\\theta\\propto N^{-1}$ under the ultrametric model, whereas independent dephasing collapses GHZ coherence exponentially in $N$. The principal result is a scaling prediction that is falsifiable by measuring nested‑event statistics; if the prediction fails, the ultrametric hypothesis is rejected. The analysis connects the model to $p$‑adic mathematical‑physics literature and documents the conventions adopted.",
  "date": "2026-10-11",
  "first_published": "2026-10-11",
  "last_revised": "2026-10-21T08:31:16Z",
  "modified": "2026-10-21T08:31:16Z",
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  "url": "https://papers.qnfo.org/papers/ultrametric-p-adic-noise-structure-and-heisenberg-limited-quantum-metrology-erro",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Ultrametric p‑Adic Noise Structure and Heisenberg‑Limited Quantum Metrology: Error Resilience by Geometry, Not Overhead"
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        "description": "Quantum metrology seeks phase sensitivities that surpass the standard quantum limit (SQL) Δθ∝ N^-1/2, aiming for the Heisenberg limit Δθ∝ N^-1. Decoherence typically destroys this advantage, and active error‑correction incurs prohibitive overhead for near‑term sensors. This work addresses the gap that no passive noise‑geometry mechanism has been quantitatively analyzed for metrological scaling. By modeling decoherence as a p‑adic hierarchical stochastic process on an ultrametric lattice, the strong triangle inequality forces error events on nested clusters to be nested rather than independent. For a hierarchy of L levels with level‑k error weight ε_k=ε₀p^-k, the ultrametric error probability is P_U=ε₀p^-L, while the Archimedean independent‑event union bound is P_A=ε₀frac1-p^-(L+1)1-p^-1. With p=2, L=10, ε₀=10^-2 the ultrametric probability is 9.765625×10^-6 versus an Archimedean bound of 1.9990234375×10^-2, a ratio of exactly $2047$. Propagating these probabilities to the quantum Fisher information (QFI) of an N‑probe GHZ state with L=log₂N yields Heisenberg scaling Δθ∝ N^-1 under the ultrametric model, whereas independent dephasing collapses GHZ coherence exponentially in N. The principal result is a scaling prediction that is falsifiable by measuring nested‑event statistics; if the prediction fails, the ultrametric hypothesis is rejected. The analysis connects the model to p‑adic mathematical‑physics literature and documents the conventions adopted.",
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