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Ballistic Transport on the Bruhat-Tits Tree: A Quantum-Native P-adic Framework for Hierarchical Explainable AI

Rowan Brad Quni-Gudzinas

Published Feb 12, 2026• 0 citations
Publisher

Abstract

Modern artificial intelligence, particularly in high-stakes domains, is hampered by a fundamental geometric mismatch: the reliance on continuous, “flat” Euclidean spaces to represent intrinsically hierarchical data. This incongruity leads to information loss and the proliferation of opaque “black box” models (Dasgupta, 2016). This paper posits that the ultrametric geometry of p-adic numbers, which axiomatically encodes hierarchy, provides a more natural and powerful foundation for AI. We argue that combining this geometric framework with the dynamics of quantum walks offers a transformative path toward building intrinsically interpretable or “glass box” models. To validate this thesis, we introduce the Quantum-Native p-adic Neural Network (Q-PNA), a simulated architecture designed to leverage these principles. The methodology involves embedding synthetic hierarchical data into a p-adic latent space, represented by the Bruhat-Tits tree, and navigating this space using a simulated quantum walk. We conduct a rigorous comparative analysis of the structural fidelity of p-adic embeddings against both Euclidean and simplified hyperbolic alternatives, using distortion and rank correlation metrics. The p-adic embeddings preserve hierarchical structure with near-perfect fidelity (Spearman’s $\rho \approx 1.0$), significantly outperforming the highly distorted representations produced by Euclidean and hyperbolic proxy models. Critically, our simulations suggest that the quantum walk exhibits ballistic transport, enabling traversal of the latent space in a time that scales linearly with its depth, $O(D)$, a quadratic speedup over classical random walks. These findings establish that a p-adic, quantum-native approach is not merely a viable alternative but a fundamentally superior paradigm for modeling hierarchical data. This work provides a simulation-based proof-of-concept for $O(D)$ navigation on a Bruhat-Tits tree for AI and offers a concrete pathway to address critical gaps in explainable AI (XAI) by creating models where the decision-making process is a geometrically interpretable path. We propose a formal “holographic dictionary” mapping neural network concepts to their geometric counterparts, paving the way for a new generation of auditable and high-fidelity AI systems.

DOI
10.5281/zenodo.18619077

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Modern artificial intelligence, particularly in high-stakes domains, is hampered by a fundamental geometric mismatch: the reliance on continuous, “flat” Euclidean spaces to represent intrinsically hierarchical data. This incongruity leads to information loss and the proliferation of opaque “black box” models (Dasgupta, 2016). This paper posits that the ultrametric geometry of p-adic numbers, which axiomatically encodes hierarchy, provides a more natural and powerful foundation for AI. We argue that combining this geometric framework with the dynamics of quantum walks offers a transformative path toward building intrinsically interpretable or “glass box” models. To validate this thesis, we introduce the Quantum-Native p-adic Neural Network (Q-PNA), a simulated architecture designed to leverage these principles. The methodology involves embedding synthetic hierarchical data into a p-adic latent space, represented by the Bruhat-Tits tree, and navigating this space using a simulated quantum walk. We conduct a rigorous comparative analysis of the structural fidelity of p-adic embeddings against both Euclidean and simplified hyperbolic alternatives, using distortion and rank correlation metrics. The p-adic embeddings preserve hierarchical structure with near-perfect fidelity (Spearman’s ρ≈1.0\rho \approx 1.0ρ≈1.0), significantly outperforming the highly distorted representations produced by Euclidean and hyperbolic proxy models. Critically, our simulations suggest that the quantum walk exhibits ballistic transport, enabling traversal of the latent space in a time that scales linearly with its depth, O(D)O(D)O(D), a quadratic speedup over classical random walks. These findings establish that a p-adic, quantum-native approach is not merely a viable alternative but a fundamentally superior paradigm for modeling hierarchical data. This work provides a simulation-based proof-of-concept for O(D)O(D)O(D) navigation on a Bruhat-Tits tree for AI and offers a concrete pathway to address critical gaps in explainable AI (XAI) by creating models where the decision-making process is a geometrically interpretable path. We propose a formal “holographic dictionary” mapping neural network concepts to their geometric counterparts, paving the way for a new generation of auditable and high-fidelity AI systems.
The probabilistic nature of prevalent primality tests, such as Miller-Rabin, presents a fundamental limitation where certainty is achieved only asymptotically. This study proposes a deterministic alternative rooted in the non-Archimedean geometry of p-adic numbers, specifically leveraging the spectral and dynamic properties of quantum walks on Bruhat-Tits trees. By modeling prime and composite numbers as distinct topological structures—regular trees for primes and Cartesian product graphs for composites—we demonstrate that their quantum dynamic signatures are fundamentally different. Our methodology combines a rigorous construction of the graph Laplacian for these spaces with a continuous-time quantum walk simulation. Analysis reveals that while both structures exhibit a robust, non-zero spectral gap (λ2≈0.0968\lambda_2 \approx 0.0968λ2​≈0.0968), their dynamic responses diverge: quantum walks on prime trees exhibit localization, whereas walks on composite product spaces lead to near-maximal delocalization across the graph (Participation Ratio ≈11.97\approx 11.97≈11.97). This binary dynamic behavior—localization versus delocalization—serves as a deterministic signal for primality. Furthermore, multi-base triangulation across p=2,3,5p=2, 3, 5p=2,3,5 yields a predictive correlation of R2=0.998R^2 = 0.998R2=0.998, confirming the robustness of the underlying metric. These findings establish a framework for a novel class of deterministic primality tests, relying on dynamic signatures rather than static spectral properties.
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 prevailing quantum computing paradigm is fundamentally constrained by the continuous analog fragility inherent to Archimedean spaces, resulting in uncontrolled linear error accumulation that demands massive active correction overheads. To address this structural bottleneck, we propose a transition to a non-Archimedean state space modeled on the p-adic numbers and their graph-theoretic realization, the Bruhat-Tits tree. To validate this non-Archimedean architecture without physical p-adic hardware substrates, we developed a comprehensive software emulation utilizing Bounded Algorithmic Number (BAN) arithmetic logic units to process exact ultrametric valuations. We simulated the compilation of continuous logic gates into discrete tree automorphisms, mapping standard unitary operators to vertex shifts and branch permutations on a p=2 Cayley graph. Furthermore, stochastic Ohmic and burst noise injections were modeled to test the error-filtering properties of the strong triangle inequality dynamically, while van der Put Neural Networks (v-PuNNs) were deployed as read-out trackers to prevent topological distortion. Our empirical simulation results demonstrate that non-Archimedean architectures natively suppress linear error accumulation, with variance saturating flatly at local cluster boundaries, confirming passive geometric fault tolerance. These findings carry implications for the future of post-NISQ hardware, formally bridging holographic tensor network theory with applied digital emulation to outline a scalable computational substrate.
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.
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.]