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× 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)≤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≥3 for physical error rates up to
perr=0.40, while equivalent flat (Archimedean) encodings failed with logical error rates up to
0.152. The energy barrier protecting logical states scales as
Ebarrier(d)=2d, 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.
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.]
We propose a fault-tolerant quantum computing architecture—the Bruhat-Tits Quantum Processor—that exploits the ultrametric geometry of the Bruhat-Tits tree
Tp 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
X+
Z, 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% (
4.6× above surface codes at
∼10.9%), a depolarizing threshold of
pc=75.0% (
75× above surface codes at
∼1.0%), and an independent
X+
Z threshold of
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=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)) confines error clusters to disjoint subtrees, enabling parallel, local decoding in
O(slogs) time.
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), 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.
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) to
O(logN). We verify Continuous-Time Quantum Walks (CTQW) achieve ballistic transport on hierarchical graphs, with hitting times scaling as
O(D) (
R2=0.994), utilizing a radial symmetric path proxy for high-branching systems (
p≥3). Langevin simulations confirm ultrametric trapping with a time-dependent slope reduction (0.59
→ 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 persistent challenges in scaling conventional quantum computers suggest a potential misalignment in foundational assumptions. Assuming that physical reality is continuous at the lowest quantum levels forces engineers to fight against natural thermodynamic tendencies. The continuous error model demands an active correction paradigm that may be fundamentally unsustainable for large-scale computation. Alternative mathematical frameworks offer different geometric properties that could naturally suppress error accumulation. Exploring non-Archimedean geometries provides a pathway to bypass the limitations inherent in continuous state spaces. These alternative structures possess discrete topological features that fundamentally alter how errors propagate through a system. Shifting the mathematical foundation could resolve the scaling bottlenecks currently hindering quantum technology development.