{
  "id": "oai:papers.qnfo.org:ultrametric-tree-encoding-for-passive-quantum-error-confinement-support-confinem",
  "slug": "ultrametric-tree-encoding-for-passive-quantum-error-confinement-support-confinem",
  "title": "Ultrametric Tree Encoding for Passive Quantum Error Confinement: Support Confinement, Barrier Models, and the Limits of the Strong Triangle Inequality",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Scalable quantum computing is blocked by the overhead of active error correction: surface-code implementations require on the order of 10³ physical qubits per logical qubit, repeated syndrome extraction, and low-latency decoding. A recent proposal claims that encoding logical information on a regular odd-p Bruhat–Tits tree, whose distance obeys the strong triangle inequality, produces a passive error barrier under which the logical error rate vanishes at tree depth d≥ 3. The specific gap we address is that this claim conflates a metric-axiom statement with a probabilistic statement about error propagation. We formalize error propagation as paths on the tree, derive from the strong triangle inequality a support-confinement result for minimal error paths, and show that the posited exponential barrier E_barrier(d)=2^d follows from cut counting on any rooted tree and invokes the ultrametric axiom nowhere. We compute the recursive eight-fold majority-vote cascade at physical error rate p_phys=0.4 in full: p^(1)=0.1736704, p^(2)=5.5489×10^-3, p^(3)≈2.905×10^-10, with suppression factors σ(3)≈6.10×10² and σ(4)≈9.31×10⁵. We further show that a zero-error simulation report bounds the logical error probability only at p_L≤2.9957×10^-5 even assuming N=10⁵ independent trials. The practical conclusion is that ultrametric encoding yields support confinement, not proven error suppression; the claimed p_L=0 rests on unverified noise assumptions and must be tested against correlated noise before any design decision relies on it.",
  "abstract_markdown": "Scalable quantum computing is blocked by the overhead of active error correction: surface-code implementations require on the order of $10^{3}$ physical qubits per logical qubit, repeated syndrome extraction, and low-latency decoding. A recent proposal claims that encoding logical information on a regular odd-$p$ Bruhat–Tits tree, whose distance obeys the strong triangle inequality, produces a passive error barrier under which the logical error rate vanishes at tree depth $d\\ge 3$. The specific gap we address is that this claim conflates a metric-axiom statement with a probabilistic statement about error propagation. We formalize error propagation as paths on the tree, derive from the strong triangle inequality a support-confinement result for minimal error paths, and show that the posited exponential barrier $E_{\\text{barrier}}(d)=2^{d}$ follows from cut counting on any rooted tree and invokes the ultrametric axiom nowhere. We compute the recursive eight-fold majority-vote cascade at physical error rate $p_{\\text{phys}}=0.4$ in full: $p^{(1)}=0.1736704$, $p^{(2)}=5.5489\\times10^{-3}$, $p^{(3)}\\approx2.905\\times10^{-10}$, with suppression factors $\\sigma(3)\\approx6.10\\times10^{2}$ and $\\sigma(4)\\approx9.31\\times10^{5}$. We further show that a zero-error simulation report bounds the logical error probability only at $p_{L}\\le2.9957\\times10^{-5}$ even assuming $N=10^{5}$ independent trials. The practical conclusion is that ultrametric encoding yields support confinement, not proven error suppression; the claimed $p_{L}=0$ rests on unverified noise assumptions and must be tested against correlated noise before any design decision relies on it.",
  "date": "2026-11-04",
  "first_published": "2026-11-04",
  "last_revised": "2026-11-04T00:00:00Z",
  "modified": "2026-11-04T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/ultrametric-tree-encoding-for-passive-quantum-error-confinement-support-confinem",
  "license_url": "https://legal.qnfo.org/",
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  "version_url": "https://papers.qnfo.org/papers/ultrametric-tree-encoding-for-passive-quantum-error-confinement-support-confinem/v/1.0.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Ultrametric Tree Encoding for Passive Quantum Error Confinement: Support Confinement, Barrier Models, and the Limits of the Strong Triangle Inequality"
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    "publicationYear": "2026",
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        "description": "Scalable quantum computing is blocked by the overhead of active error correction: surface-code implementations require on the order of 10³ physical qubits per logical qubit, repeated syndrome extraction, and low-latency decoding. A recent proposal claims that encoding logical information on a regular odd-p Bruhat–Tits tree, whose distance obeys the strong triangle inequality, produces a passive error barrier under which the logical error rate vanishes at tree depth d≥ 3. The specific gap we address is that this claim conflates a metric-axiom statement with a probabilistic statement about error propagation. We formalize error propagation as paths on the tree, derive from the strong triangle inequality a support-confinement result for minimal error paths, and show that the posited exponential barrier E_barrier(d)=2^d follows from cut counting on any rooted tree and invokes the ultrametric axiom nowhere. We compute the recursive eight-fold majority-vote cascade at physical error rate p_phys=0.4 in full: p^(1)=0.1736704, p^(2)=5.5489×10^-3, p^(3)≈2.905×10^-10, with suppression factors σ(3)≈6.10×10² and σ(4)≈9.31×10⁵. We further show that a zero-error simulation report bounds the logical error probability only at p_L≤2.9957×10^-5 even assuming N=10⁵ independent trials. The practical conclusion is that ultrametric encoding yields support confinement, not proven error suppression; the claimed p_L=0 rests on unverified noise assumptions and must be tested against correlated noise before any design decision relies on it.",
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