{
  "id": "oai:papers.qnfo.org:ultrametric-tree-connectivity-as-a-passive-fault-tolerance-mechanism-for-quantum",
  "slug": "ultrametric-tree-connectivity-as-a-passive-fault-tolerance-mechanism-for-quantum",
  "title": "Ultrametric Tree Connectivity as a Passive Fault-Tolerance Mechanism for Quantum Processors",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Fault-tolerant quantum computation currently relies on active stabilizer codes that demand repeated syndrome extraction, low-latency classical decoding, and large ancilla overhead; we assume a surface-code overhead in the range 10²–10⁴ physical qubits per logical qubit for comparison, while [9] estimates a 10³ : 1 ratio under its own assumptions. A passive protection mechanism that exploits hardware geometry rather than active decoding cycles would reduce this overhead, but no established design principle converts connectivity geometry directly into an error barrier. We investigate the hypothesis that arranging qubits on a Bruhat–Tits (ultrametric) tree confines correlated errors within subtrees through the strong triangle inequality. Using a stochastic depolarizing model with per-edge error probability ε, we derive an exponential barrier B(d)=2^d at tree depth d, the logical error bound p_L(d)=ε^B(d), and the suppression factor S(d)=ε/p_L(d). For ε=0.4 we compute p_L(3)=6.5536×10^-4 and p_L(4)=4.2950×10^-7, giving suppression factors of 6.10×10² and 9.31×10⁵. Term-by-term evaluation of a recursive 8-leaf majority vote yields failure probabilities $0.1736704$, 5.55×10^-3, and 2.905×10^-10 at successive levels, matching the values reported by prior barrier analyses. Ball-code size formulas give $5$, $17$, and $53$ qudits for depths $1$–$3$ on a ternary tree. At ε=0.4, the model yields suppression factors of 6.10×10² at depth 3 and 9.31×10⁵ at depth 4, and it defines a five-atom prototype experiment that would confirm or falsify the barrier model.",
  "abstract_markdown": "Fault-tolerant quantum computation currently relies on active stabilizer codes that demand repeated syndrome extraction, low-latency classical decoding, and large ancilla overhead; we assume a surface-code overhead in the range $10^{2}$–$10^{4}$ physical qubits per logical qubit for comparison, while [9] estimates a $10^{3}\\!:\\!1$ ratio under its own assumptions. A passive protection mechanism that exploits hardware geometry rather than active decoding cycles would reduce this overhead, but no established design principle converts connectivity geometry directly into an error barrier. We investigate the hypothesis that arranging qubits on a Bruhat–Tits (ultrametric) tree confines correlated errors within subtrees through the strong triangle inequality. Using a stochastic depolarizing model with per-edge error probability $\\varepsilon$, we derive an exponential barrier $B(d)=2^{d}$ at tree depth $d$, the logical error bound $p_{L}(d)=\\varepsilon^{B(d)}$, and the suppression factor $S(d)=\\varepsilon/p_{L}(d)$. For $\\varepsilon=0.4$ we compute $p_{L}(3)=6.5536\\times10^{-4}$ and $p_{L}(4)=4.2950\\times10^{-7}$, giving suppression factors of $6.10\\times10^{2}$ and $9.31\\times10^{5}$. Term-by-term evaluation of a recursive 8-leaf majority vote yields failure probabilities $0.1736704$, $5.55\\times10^{-3}$, and $2.905\\times10^{-10}$ at successive levels, matching the values reported by prior barrier analyses. Ball-code size formulas give $5$, $17$, and $53$ qudits for depths $1$–$3$ on a ternary tree. At $\\varepsilon=0.4$, the model yields suppression factors of $6.10\\times10^{2}$ at depth 3 and $9.31\\times10^{5}$ at depth 4, and it defines a five-atom prototype experiment that would confirm or falsify the barrier model.",
  "date": "2026-11-07",
  "first_published": "2026-11-07",
  "last_revised": "2026-11-07T00:00:00Z",
  "modified": "2026-11-07T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/ultrametric-tree-connectivity-as-a-passive-fault-tolerance-mechanism-for-quantum",
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  "version_url": "https://papers.qnfo.org/papers/ultrametric-tree-connectivity-as-a-passive-fault-tolerance-mechanism-for-quantum/v/1.0.0",
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    "creators": [
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Ultrametric Tree Connectivity as a Passive Fault-Tolerance Mechanism for Quantum Processors"
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    "publicationYear": "2026",
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        "description": "Fault-tolerant quantum computation currently relies on active stabilizer codes that demand repeated syndrome extraction, low-latency classical decoding, and large ancilla overhead; we assume a surface-code overhead in the range 10²–10⁴ physical qubits per logical qubit for comparison, while [9] estimates a 10³ : 1 ratio under its own assumptions. A passive protection mechanism that exploits hardware geometry rather than active decoding cycles would reduce this overhead, but no established design principle converts connectivity geometry directly into an error barrier. We investigate the hypothesis that arranging qubits on a Bruhat–Tits (ultrametric) tree confines correlated errors within subtrees through the strong triangle inequality. Using a stochastic depolarizing model with per-edge error probability ε, we derive an exponential barrier B(d)=2^d at tree depth d, the logical error bound p_L(d)=ε^B(d), and the suppression factor S(d)=ε/p_L(d). For ε=0.4 we compute p_L(3)=6.5536×10^-4 and p_L(4)=4.2950×10^-7, giving suppression factors of 6.10×10² and 9.31×10⁵. Term-by-term evaluation of a recursive 8-leaf majority vote yields failure probabilities $0.1736704$, 5.55×10^-3, and 2.905×10^-10 at successive levels, matching the values reported by prior barrier analyses. Ball-code size formulas give $5$, $17$, and $53$ qudits for depths $1$–$3$ on a ternary tree. At ε=0.4, the model yields suppression factors of 6.10×10² at depth 3 and 9.31×10⁵ at depth 4, and it defines a five-atom prototype experiment that would confirm or falsify the barrier model.",
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