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

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.