{
  "id": "oai:papers.qnfo.org:ultrametric-geometry-as-passive-quantum-error-correction-bruhat-tits-qudit-codes",
  "slug": "ultrametric-geometry-as-passive-quantum-error-correction-bruhat-tits-qudit-codes",
  "title": "Ultrametric Geometry as Passive Quantum Error Correction: Bruhat–Tits Qudit Codes and the Non-Archimedean Bound",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "We formalize a proposal for passive quantum error correction (QEC) in which logical qudits are encoded on the vertices of a ball of the Bruhat–Tits tree T_p+1, the incidence geometry of the p‑adic numbers Q_p, with Hecke‑type operators acting as the natural gate set. The non‑Archimedean inequality \\|x+y\\|_p≤max(\\|x\\|_p,\\|y\\|_p) forces the tree’s balls to be either nested or disjoint, so localized errors confined to one ball cannot accumulate coherently across distinct branches. We prove a nesting lemma, use it to show that every error set of weight ≤ t on a protected ball is correctable under the Knill–Laflamme conditions, and derive the code parameters n_t=1+(p+1)(p^t-1)/(p-1) and d=2t+1. For p=3, t=1 this gives a distance‑3 code on n₁=5 qudits with rate 1/5. Under independent depolarizing noise at physical error rate ε=10^-3 we compute, with full arithmetic, a passive logical failure probability P_pass≈9.98×10^-6, a suppression factor of ≈100.2 over an unencoded qudit, and, as an explicitly labeled projection, a ≈4.08‑fold advantage over an active distance‑3 baseline with noisy syndrome extraction. All figures are analytic projections under stated assumptions; limitations and falsification criteria are discussed.",
  "abstract_markdown": "We formalize a proposal for passive quantum error correction (QEC) in which logical qudits are encoded on the vertices of a ball of the Bruhat–Tits tree $T_{p+1}$, the incidence geometry of the $p$‑adic numbers $\\mathbb{Q}_p$, with Hecke‑type operators acting as the natural gate set. The non‑Archimedean inequality $\\|x+y\\|_p\\le\\max(\\|x\\|_p,\\|y\\|_p)$ forces the tree’s balls to be either nested or disjoint, so localized errors confined to one ball cannot accumulate coherently across distinct branches. We prove a nesting lemma, use it to show that every error set of weight $\\le t$ on a protected ball is correctable under the Knill–Laflamme conditions, and derive the code parameters $n_t=1+(p+1)(p^{t}-1)/(p-1)$ and $d=2t+1$. For $p=3$, $t=1$ this gives a distance‑3 code on $n_1=5$ qudits with rate $1/5$. Under independent depolarizing noise at physical error rate $\\varepsilon=10^{-3}$ we compute, with full arithmetic, a passive logical failure probability $P_{\\mathrm{pass}}\\approx9.98\\times10^{-6}$, a suppression factor of $\\approx100.2$ over an unencoded qudit, and, as an explicitly labeled projection, a $\\approx4.08$‑fold advantage over an active distance‑3 baseline with noisy syndrome extraction. All figures are analytic projections under stated assumptions; limitations and falsification criteria are discussed.",
  "date": "2026-10-11",
  "first_published": "2026-10-11",
  "last_revised": "2026-10-21T05:31:08Z",
  "modified": "2026-10-21T05:31:08Z",
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  "doi": null,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/ultrametric-geometry-as-passive-quantum-error-correction-bruhat-tits-qudit-codes",
  "license_url": "https://legal.qnfo.org/",
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  "version_url": "https://papers.qnfo.org/papers/ultrametric-geometry-as-passive-quantum-error-correction-bruhat-tits-qudit-codes/v/2.0.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Ultrametric Geometry as Passive Quantum Error Correction: Bruhat–Tits Qudit Codes and the Non-Archimedean Bound"
      }
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    "publicationYear": "2026",
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        "date": "2026-10-11",
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        "description": "We formalize a proposal for passive quantum error correction (QEC) in which logical qudits are encoded on the vertices of a ball of the Bruhat–Tits tree T_p+1, the incidence geometry of the p‑adic numbers Q_p, with Hecke‑type operators acting as the natural gate set. The non‑Archimedean inequality \\|x+y\\|_p≤max(\\|x\\|_p,\\|y\\|_p) forces the tree’s balls to be either nested or disjoint, so localized errors confined to one ball cannot accumulate coherently across distinct branches. We prove a nesting lemma, use it to show that every error set of weight ≤ t on a protected ball is correctable under the Knill–Laflamme conditions, and derive the code parameters n_t=1+(p+1)(p^t-1)/(p-1) and d=2t+1. For p=3, t=1 this gives a distance‑3 code on n₁=5 qudits with rate 1/5. Under independent depolarizing noise at physical error rate ε=10^-3 we compute, with full arithmetic, a passive logical failure probability P_pass≈9.98×10^-6, a suppression factor of ≈100.2 over an unencoded qudit, and, as an explicitly labeled projection, a ≈4.08‑fold advantage over an active distance‑3 baseline with noisy syndrome extraction. All figures are analytic projections under stated assumptions; limitations and falsification criteria are discussed.",
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