Atomic Orbital Machine Learning: A Hilbert-Space Learning Framework via Schrödinger Potential Dynamics and Quantum Density Matrix Interference

Classical machine learning models operate in finite-dimensional Euclidean feature spaces and rely on stacked parametric transformations to approximate nonlinear decision boundaries—a strategy that often incurs high computational cost, susceptibility to the curse of dimensionality, and limited interpretability. We introduce Atomic Orbital Machine Learning (AOML), a paradigm that recasts the learning problem in continuous Hilbert spaces by encoding data instances as quantum wavefunctions within an atomic potential field. In this formulation, classification boundaries emerge as nodal surfaces of destructive wave interference, regression is realized through continuous potential expectation values, and unsupervised clustering arises from the spectral decomposition of quantum density matrices. AOML operates through two dual inference pathways: (i) Schrödinger Potential Dynamics, which assigns labels by minimizing Hamiltonian energy expectation values across class-specific potential wells, and (ii) Quantum Density Matrix Statistics, which performs classification via quantum fidelity maximization using mixed-state density operators. Both paths yield O(C·K) inference complexity—entirely independent of training set size—and are fully differentiable, enabling end-to-end training via gradient-based optimization. The framework unifies Classification, Regression, and Clustering under a single theoretical roof while offering inherent noise robustness through Gaussian wavepacket smoothing and physically grounded interpretability via energy spectra and orbital visualizations. We present the complete mathematical formalism, algorithmic design, and loss formulations, and validate the framework empirically across eight benchmark datasets spanning classification, regression, and clustering tasks. Results confirm the theoretical prediction of M-independent inference latency (AOML remains flat at 2.1–3.1 ms while SVC grows 5.5×) and demonstrate competitive accuracy with classical baselines on low-to-moderate-dimensional datasets, along with superior relative noise robustness.