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KAN-DAM Quantum Computing on the 861-Node Spectral Manifold: Algorithmic Architecture, Non-Linear Dynamics, and Universal Hardware Realization

Michael Aksman

Published Nov 1, 2026• 0 citations
Publisher

Abstract

The KAN-DAM computing architecture establishes a deterministic quantum-computational model hosted on a finite 861-node spectral manifold, replacing isolated physical qubits and discrete gate sequences with globally distributed, non-local state representations. Information processing advances through an alternating cycle of linear functional mode mixing, non-linear dense associative coupling, and immediate geometric projection back onto the constrained manifold. The system integrates global Chebyshev spectral polynomial expansions with localized adaptive spline representations, coupling macro-level coherence with localized gradient absorption to eliminate oscillatory instabilities. The non-linear stage generates many-body interactions and effective entanglement in a single transformation without requiring deep sequences of two-qubit operations. Error suppression operates as an automatic geometric filter rather than an active algorithmic routine: perturbations that drift outside the rigid manifold constraints are discarded by the projection step, eliminating the need for ancilla qubits or syndrome measurements. This design circumvents the exponential entanglement scaling limits of classical tensor networks and the critical gap-closing slowdowns of adiabatic quantum annealing, providing an inherently stable platform that can be simulated classically or implemented across photonic, superconducting, and cold-atom hardware.

DOI
10.5281/zenodo.23850577

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The KAN-DAM computing architecture establishes a deterministic quantum-computational model hosted on a finite 861-node spectral manifold, replacing isolated physical qubits and discrete gate sequences with globally distributed, non-local state representations. Information processing advances through an alternating cycle of linear functional mode mixing, non-linear dense associative coupling, and immediate geometric projection back onto the constrained manifold. The system integrates global Chebyshev spectral polynomial expansions with localized adaptive spline representations, coupling macro-level coherence with localized gradient absorption to eliminate oscillatory instabilities. The non-linear stage generates many-body interactions and effective entanglement in a single transformation without requiring deep sequences of two-qubit operations. Error suppression operates as an automatic geometric filter rather than an active algorithmic routine: perturbations that drift outside the rigid manifold constraints are discarded by the projection step, eliminating the need for ancilla qubits or syndrome measurements. This design circumvents the exponential entanglement scaling limits of classical tensor networks and the critical gap-closing slowdowns of adiabatic quantum annealing, providing an inherently stable platform that can be simulated classically or implemented across photonic, superconducting, and cold-atom hardware.
This work introduces a new paradigm for quantum computation, here termed Geometric Quantum Computing (GQC), in which entangling gates arise not from local physical interactions but from global geometric modes encoded in a Unified Effective Average Action (UEAA). Building on recent optical experiments demonstrating entanglement without interaction—notably the path-identity schemes of Krenn et al. (2017) and Wang et al. (2024)—the paper shows that such processes are naturally described by a bilocal kernel in the UEAA, activated by geometric constraints rather than Hamiltonian coupling. A scalar quantity termed entanglement curvature is derived from the bilocal kernel and shown to be equivalent to concurrence in the single-photon subspace. This suggests a method for designing quantum processors in which entanglement is generated through engineered geometric overlap of mode functions, potentially reducing resource requirements for distributed quantum networks. The framework extends techniques from the IGM–RG program (Medina Albarenga, 2025) to quantum information science and offers a unified geometric viewpoint on entanglement generation, error mechanisms, and network architecture.
We present a rigorous head-to-head analytical comparison between the KAN-DAM (KolmogorovArnold Network + Dense Associative Memory) architecture operating on a xed nite spectral manifold (N = 861) and conventional tensor-network (TN) methods. Tensor networks suer exponential scaling with bond dimension due to volume-law entanglement generation, and their exact contraction in 2D/3D is strictly #P-hard, forcing systematic topological destruction via Singular Value Decomposition (SVD) truncation. Conversely, KAN-DAMmaintains a xed dimensionality N, achieves a bounded per-step complexity of O(N2) and total cost O(N2T), and suppresses non-physical errors globally via Dense Associative Memory geometric projection rather than truncation. The Kolmogorov-Arnold representation guarantees dense functional expressivity without requiring explicit, localized entanglement management. This establishes KAN-DAM as a bounded-cost, geometrically stabilized computational paradigm ideally suited for non-perturbative problems whose physics is constrained to a low-dimensional topological surface.
Quantum state non-linear entanglement computation is a burgeoning field with immense potential for advancing computational capabilities. This paper explores a novel computational model predicated on quantum entanglement, aiming to overcome the limitations of classical computation in processing complex non-linear quantum states. The core mechanism centers around leveraging entanglement's unique properties to facilitate efficient and highly complex operations on quantum states. This research focuses on the design of a quantum computing framework that utilizes entanglement to achieve computational speedups and offers a fundamentally different approach to problem-solving. The paper details the model's architecture, explores its potential for specific algorithms, and discusses its implications for future advancements in quantum computation.
We present an explicit algorithmic realization of quantum computation on a finitespectral manifold of dimension N = 861 using the KAN–DAM architecture. Logical states areencoded nonlocally across all modes, and evolution proceeds through repeated nonlineartransformations followed by a deterministic projection onto the constrained manifold. Thismechanism supplies intrinsic error suppression and state-independent dynamics withoutrequiring external error-correcting codes or precise gate calibration. We detail the update rule,prove universality through functional approximation and nonlinear coupling, compare theapproach to tensor-network methods and adiabatic quantum computing, and provide a minimalsimulation protocol that can be implemented immediately on classical hardware. The result is acomputational paradigm in which logic, error correction, and thermodynamic consistency allemerge naturally from projection onto a finite-capacity spectral structure.
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The DSM-861 error-correction mechanism—defined as the instantaneous orthogonal projection Π861 onto the Heegner-rigid 861-node vorton lattice—is systematically applied and mathematically resolved across three disparate physical domains. (I) Quantum Memories: We unify the DSM-861 projection-enforced admissibility with the 2026 3D sheaf-code passive quantum memory (Balasubramanian et al., arXiv:2605.10943) and the classic 4D loop toric code (Dennis et al., 2002), demonstrating how arithmetic rigidity completely bypasses active decoder overhead to achieve hardware-level zero-drift. (II) Biological Error Catastrophe: We resolve the Meyer-Axe paradox of protein folding by quantifying the Spectral Admissibility Criterion (SAC). We establish the exact thermodynamic threshold linking admissibility gaps to folding free energy, formulate the topological restoration force, and derive the biological longevity limit as a deterministic consequence of KolmogorovArnold-Moser (KAM) stability breakdown. (III) KAN-DAM Quantum Computation: We prove the computational universality of the KAN-DAM framework, encoding states nonlocally across all 861 modes. We demonstrate that the system avoids the exponential scaling of volume-law entanglement by utilizing the Π861 projection as a continuous physical disentangler, bounding classical simulation complexity strictly to polynomial time O(N2log(1/)). This unified paradigm requires no external error-correcting codes, replacingprobabilistic syndrome measurements with deterministic geometric necessity.