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Spectral Dynamics and Thermodynamic Stability in the Arithmetic Quantum Framework: Reconciling Ultrametric Geometry with Macroscopic Information Storage

Rowan Brad Quni-Gudzinas

Published Feb 14, 2026• 0 citations
Publisher

Abstract

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 $O(N \cdot d^2)$ to $O(\log N)$. We verify Continuous-Time Quantum Walks (CTQW) achieve ballistic transport on hierarchical graphs, with hitting times scaling as $O(D)$ ($R^2 = 0.994$), utilizing a radial symmetric path proxy for high-branching systems ($p \ge 3$). 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.

DOI
10.5281/zenodo.18641725

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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 scalability of Topological Quantum Computing (TQC) is currently impeded by a “Thermodynamic Wall,” where the entropy generation from active error correction cycles scales exponentially with logical qubit count. This paper proposes a paradigm shift from active gate synthesis to passive topological relaxation, grounded in the Quni-Gudzinas hypothesis (2025). By establishing a rigorous functorial isomorphism between the inverse limit of abelian anyonic braid groups and the p-adic solenoid (Σp\Sigma_pΣp​), we demonstrate that the vacuum structure of specific strain-engineered materials can encode topological quantum information. We derive a 2D Hamiltonian for a hierarchical “synthetic vacuum” where relaxation dynamics follow an ultrametric trajectory, effectively freezing the system into a protected topological sector without external intervention. While this architecture is limited to abelian topological sectors and thus functions primarily as a quantum memory rather than a universal processor, simulation results indicate that it reduces thermodynamic overhead by orders of magnitude compared to surface code implementations, offering a viable path toward macroscopic fault tolerance for storage.
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 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 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.]