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Computational Validation of Ultrametric Error Confinement in Bruhat–Tits Tree Quantum Circuits

Rowan Brad Quni-Gudzinas

Published May 12, 2026• 0 citations
Publisher

Abstract

The standard mathematical foundation of quantum computing—Archimedean (Euclidean) geometry—assumes that space is continuous, distances add linearly, and small errors can accumulate into large ones. This assumption creates a fundamental bottleneck: active quantum error correction generates heat that scales faster than cryogenic cooling capacity, producing a $20{,}000\times$ gap between required and available cooling at millikelvin temperatures. We propose an alternative: encoding quantum states on Bruhat–Tits trees, the geometric realization of $p$-adic (ultrametric) spaces, where the strong triangle inequality $d(x,z) \leq \max\{d(x,y), d(y,z)\}$ geometrically confines errors to their local branches. We present a computational validation suite (classical simulation) demonstrating this mechanism. Our results show that tree-encoded circuits produced zero observed logical errors in 500 trials at depths $d \geq 3$ for physical error rates up to $p_{\text{err}} = 0.40$, while equivalent flat (Archimedean) encodings failed with logical error rates up to $0.152$. The energy barrier protecting logical states scales as $E_{\text{barrier}}(d) = 2^d$, confirmed exhaustively for $d = 2, 3$ and computed analytically to $d = 10$. The strong triangle inequality is verified with zero violations in $15{,}000$ random trials across primes $p = 2, 3, 5$. These results provide computational evidence for a geometric threshold theorem in ultrametric quantum error correction and suggest a path toward passive fault tolerance—error suppression as a property of hardware geometry rather than active measurement protocols. The simulation uses classical bits with majority-vote decoding; full quantum validation with superposition and entanglement remains future work.

DOI
10.5281/zenodo.20134944

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The standard mathematical foundation of quantum computing—Archimedean (Euclidean) geometry—assumes that space is continuous, distances add linearly, and small errors can accumulate into large ones. This assumption creates a fundamental bottleneck: active quantum error correction generates heat that scales faster than cryogenic cooling capacity, producing a 20,000×20{,}000\times20,000× gap between required and available cooling at millikelvin temperatures. We propose an alternative: encoding quantum states on Bruhat–Tits trees, the geometric realization of ppp-adic (ultrametric) spaces, where the strong triangle inequality d(x,z)≤max⁡{d(x,y),d(y,z)}d(x,z) \leq \max\{d(x,y), d(y,z)\}d(x,z)≤max{d(x,y),d(y,z)} geometrically confines errors to their local branches. We present a computational validation suite (classical simulation) demonstrating this mechanism. Our results show that tree-encoded circuits produced zero observed logical errors in 500 trials at depths d≥3d \geq 3d≥3 for physical error rates up to perr=0.40p_{\text{err}} = 0.40perr​=0.40, while equivalent flat (Archimedean) encodings failed with logical error rates up to 0.1520.1520.152. The energy barrier protecting logical states scales as Ebarrier(d)=2dE_{\text{barrier}}(d) = 2^dEbarrier​(d)=2d, confirmed exhaustively for d=2,3d = 2, 3d=2,3 and computed analytically to d=10d = 10d=10. The strong triangle inequality is verified with zero violations in 15,00015{,}00015,000 random trials across primes p=2,3,5p = 2, 3, 5p=2,3,5. These results provide computational evidence for a geometric threshold theorem in ultrametric quantum error correction and suggest a path toward passive fault tolerance—error suppression as a property of hardware geometry rather than active measurement protocols. The simulation uses classical bits with majority-vote decoding; full quantum validation with superposition and entanglement remains future work.
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We propose a fault-tolerant quantum computing architecture—the Bruhat-Tits Quantum Processor—that exploits the ultrametric geometry of the Bruhat-Tits tree Tp\mathcal{T}_pTp​ to achieve quantum error correction thresholds substantially exceeding those of conventional Euclidean-locality codes. The architecture uses holographic perfect-tensor codes on a truncated Bruhat-Tits tree, tree-automorphism logical gates, and fractal-multiplexed readout. We compute bit-flip, depolarizing, independent XXX+ZZZ, gate-error, and magic-state distillation thresholds via analytical recursion, exact Gottesman-Knill stabilizer simulation, and Monte Carlo methods. The tree code achieves a bit-flip threshold of pc=50.0%p_c = 50.0\%pc​=50.0% (4.6×4.6\times4.6× above surface codes at ∼10.9%\sim 10.9\%∼10.9%), a depolarizing threshold of pc=75.0%p_c = 75.0\%pc​=75.0% (75×75\times75× above surface codes at ∼1.0%\sim 1.0\%∼1.0%), and an independent XXX+ZZZ threshold of pc=17.30%p_c = 17.30\%pc​=17.30%. We benchmark the tree code against the surface code at matched physical qubit counts using Monte Carlo simulation with Minimum-Weight Perfect Matching (MWPM) decoding. A logical-level demonstration of Shor’s algorithm for N=15N = 15N=15 confirms algorithmic completeness and establishes physical resource requirements. All quantitative claims are verified by self-contained Python simulations. We identify open problems—including perfect tensor existence, tree automorphism gate generation, and physical noise modeling—as directions for future investigation. The architecture’s defining innovation is the use of ultrametricity as a quantum error correction resource: the strong triangle inequality d(x,z)=max⁡(d(x,y),d(y,z))d(x,z) = \max(d(x,y), d(y,z))d(x,z)=max(d(x,y),d(y,z)) confines error clusters to disjoint subtrees, enabling parallel, local decoding in O(slog⁡s)O(s \log s)O(slogs) time.
The Bruhat-Tits tree T_p is proposed as a geometric substrate for quantum error-correcting codes. The defining property of ultrametric geometry replaces continuous error accumulation with discrete redundancy steps. This topological reframing connects p-adic mathematical physics, quantum error correction theory, and quantum Darwinism. The central conjecture is that the redundancy-fidelity tradeoff, parameterized on a real quantum processor, may exhibit a staircase structure (ultrametric regime) rather than a smooth curve (Archimedean limit), and that this structure — if detected — would constitute a new QEC design parameter. An experimental protocol is presented for discriminating between these regimes on existing quantum processors. All claims are graded with certainty labels; the paper acknowledges that the experimental program is the path to evidence. [v0.2 — literature search completed 2026-08-06: prior holographic-code work (Marcolli 2018) acknowledged; novelty claim refined to device-level QEC with staircase detection.]
The scalability of fault-tolerant quantum computing is currently obstructed by a “Thermodynamic Wall,” where the entropy generated by active error correction (QEC) cycles scales exponentially with the number of logical qubits. This research addresses the crisis by proposing a transition from active, entropy-driven maintenance to passive, geometry-driven protection within the Arithmetic Quantum Framework. We establish a rigorous isomorphism between the p-adic ultrametric geometry of the Bruhat-Tits tree and the physical requirements for macroscopic information storage. Utilizing p-adic statistical field theory (SFT), we model the “bulk” latent space as a self-correcting manifold. Our results demonstrate that p-adic suppression reduces entropy scaling from O(N⋅d2)O(N \cdot d^2)O(N⋅d2) to O(log⁡N)O(\log N)O(logN). We verify Continuous-Time Quantum Walks (CTQW) achieve ballistic transport on hierarchical graphs, with hitting times scaling as O(D)O(D)O(D) (R2=0.994R^2 = 0.994R2=0.994), utilizing a radial symmetric path proxy for high-branching systems (p≥3p \ge 3p≥3). Langevin simulations confirm ultrametric trapping with a time-dependent slope reduction (0.59 →\to→ 0.30), proving asymptotic logarithmic “freezing.” We acknowledge a 10nm ultraviolet cutoff in physical strain engineering. This work provides a blueprint for sustainable macroscopic quantum storage and intrinsically interpretable AI.
The Bruhat-Tits tree provides an ultrametric substrate for quantum error-correcting codes whose information topology differs from the Archimedean case in ways that may revise the redundancy-fidelity tradeoff established for quantum Darwinism. Two limiting regimes are identified: the large-p limit, in which the ultrametric bound converges to the Archimedean bound, and the deep ultrametric regime of small primes, in which discrete redundancy steps create windows where quantum error correction and classical objectivity may coexist. An experimental protocol is proposed to detect the staircase structure on real quantum processors. All claims are graded with epistemic certainty labels and subjected to an internal red-team audit. Licensed under QNFO Unified License Agreement (QNFO-ULA).
Quantum computing is stalled. Despite decades of investment, no quantum computer has solved a problem that a classical computer cannot. The standard explanation — that we need better error correction — assumes the mathematical framework itself is correct. This document argues that the framework is not correct. The assumption that quantum state space is a continuous manifold — inherited from classical physics without scrutiny — may be the root cause of the field's stagnation. Replacing the continuous manifold with an ultrametric (tree-based) geometry provides passive fault tolerance: errors are geometrically confined rather than actively corrected. This document presents the mathematical foundations, computational validation results, and a set of pre-registered falsifiable predictions. If ultrametric encoding produces the predicted error suppression, it offers a path to fault-tolerant quantum computing at 4 K with dramatically reduced qubit overhead. If it does not, the hypothesis is falsified. Either outcome advances the field.