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  "slug": "ultrametric-quantum-error-correction-on-bruhat-tits-trees-energy-barriers-classi",
  "title": "Ultrametric Quantum Error Correction on Bruhat–Tits Trees: Energy Barriers, Classical Evidence, and the Quantization Question",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Passive fault tolerance — protection of logical qubits by encoding geometry alone, without repeated syndrome measurement — would remove the dominant overhead in scalable quantum computing if it could be made to work for quantum information rather than only for classical bits. We study a conjectured scheme in which logical qubits are encoded on a Bruhat–Tits tree, the infinite homogeneous tree of a non-Archimedean local field whose metric satisfies the strong (ultrametric) triangle inequality. The conjecture states that the energy barrier against a logical error grows as E_barrier(d)=2^d in tree depth d; the supporting evidence is a classical simulation report of zero logical errors at a physical error rate p=0.4 for depth d≥3 under majority-vote decoding. The gap we address is that this evidence is entirely classical: whether the strong triangle inequality confines quantum errors under superposition, entanglement, and coherent noise is open. We formalize the encoding, derive the barrier conjecture from the ultrametric cut structure, and compute explicitly what the classical evidence establishes: one level of eight-fold majority voting maps p=0.4 to a failure probability of $0.1736704$, two concatenated levels give ≈5.55×10^-3, and three levels give ≈2.905×10^-10, confirming a genuine recursive threshold mechanism; the zero-error report itself bounds the logical error probability only at p_L≤2.9957×10^-5 even for N=10⁵ trials. We derive a thermal-activation projection p_L≤ e^-2^d under a stated barrier model, state the quantization gap as an explicit falsifiable hypothesis, and specify the quantum Monte Carlo and tensor-network tests that would resolve it. If the conjecture holds, syndrome-extraction hardware could be replaced by a static energy landscape.",
  "abstract_markdown": "Passive fault tolerance — protection of logical qubits by encoding geometry alone, without repeated syndrome measurement — would remove the dominant overhead in scalable quantum computing if it could be made to work for quantum information rather than only for classical bits. We study a conjectured scheme in which logical qubits are encoded on a Bruhat–Tits tree, the infinite homogeneous tree of a non-Archimedean local field whose metric satisfies the strong (ultrametric) triangle inequality. The conjecture states that the energy barrier against a logical error grows as $E_{\\text{barrier}}(d)=2^{d}$ in tree depth $d$; the supporting evidence is a classical simulation report of zero logical errors at a physical error rate $p=0.4$ for depth $d\\geq3$ under majority-vote decoding. The gap we address is that this evidence is entirely classical: whether the strong triangle inequality confines quantum errors under superposition, entanglement, and coherent noise is open. We formalize the encoding, derive the barrier conjecture from the ultrametric cut structure, and compute explicitly what the classical evidence establishes: one level of eight-fold majority voting maps $p=0.4$ to a failure probability of $0.1736704$, two concatenated levels give $\\approx5.55\\times10^{-3}$, and three levels give $\\approx2.905\\times10^{-10}$, confirming a genuine recursive threshold mechanism; the zero-error report itself bounds the logical error probability only at $p_{L}\\leq2.9957\\times10^{-5}$ even for $N=10^{5}$ trials. We derive a thermal-activation projection $p_{L}\\leq e^{-2^{d}}$ under a stated barrier model, state the quantization gap as an explicit falsifiable hypothesis, and specify the quantum Monte Carlo and tensor-network tests that would resolve it. If the conjecture holds, syndrome-extraction hardware could be replaced by a static energy landscape.",
  "date": "2026-10-12",
  "first_published": "2026-10-12",
  "last_revised": "2026-10-27T07:41:35Z",
  "modified": "2026-10-27T07:41:35Z",
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  "url": "https://papers.qnfo.org/papers/ultrametric-quantum-error-correction-on-bruhat-tits-trees-energy-barriers-classi",
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        "name": "Rowan Brad Quni-Gudzinas",
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        "title": "Ultrametric Quantum Error Correction on Bruhat–Tits Trees: Energy Barriers, Classical Evidence, and the Quantization Question"
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        "description": "Passive fault tolerance — protection of logical qubits by encoding geometry alone, without repeated syndrome measurement — would remove the dominant overhead in scalable quantum computing if it could be made to work for quantum information rather than only for classical bits. We study a conjectured scheme in which logical qubits are encoded on a Bruhat–Tits tree, the infinite homogeneous tree of a non-Archimedean local field whose metric satisfies the strong (ultrametric) triangle inequality. The conjecture states that the energy barrier against a logical error grows as E_barrier(d)=2^d in tree depth d; the supporting evidence is a classical simulation report of zero logical errors at a physical error rate p=0.4 for depth d≥3 under majority-vote decoding. The gap we address is that this evidence is entirely classical: whether the strong triangle inequality confines quantum errors under superposition, entanglement, and coherent noise is open. We formalize the encoding, derive the barrier conjecture from the ultrametric cut structure, and compute explicitly what the classical evidence establishes: one level of eight-fold majority voting maps p=0.4 to a failure probability of $0.1736704$, two concatenated levels give ≈5.55×10^-3, and three levels give ≈2.905×10^-10, confirming a genuine recursive threshold mechanism; the zero-error report itself bounds the logical error probability only at p_L≤2.9957×10^-5 even for N=10⁵ trials. We derive a thermal-activation projection p_L≤ e^-2^d under a stated barrier model, state the quantization gap as an explicit falsifiable hypothesis, and specify the quantum Monte Carlo and tensor-network tests that would resolve it. If the conjecture holds, syndrome-extraction hardware could be replaced by a static energy landscape.",
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