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Ultrametric Information Geometry: From p-Adic Spaces to Quantum Error Correction

QNFO Research

Published Jul 5, 2026• 0 citations
Publisher

Abstract

Ultrametric spaces provide a natural geometry for systems organized hierarchically. This paper synthesizes research connecting ultrametric geometry to quantum information theory: Murtagh's data embedding, Heydeman's holographic QEC on Bruhat-Tits trees, Palmer's p-adic quantum foundations, and Aniello's p-adic Hilbert space formalism. QNFO contributions: D=4 embedding theorem, adelic QEC, CMB analysis.

DOI
10.5281/zenodo.21204046

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Ultrametric spaces provide a natural geometry for systems organized hierarchically. This paper synthesizes research connecting ultrametric geometry to quantum information theory: Murtagh's data embedding, Heydeman's holographic QEC on Bruhat-Tits trees, Palmer's p-adic quantum foundations, and Aniello's p-adic Hilbert space formalism. QNFO contributions: D=4 embedding theorem, adelic QEC, CMB analysis.
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 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).
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.