Quantum Harmonic Oscillator-Inspired Energy-Based Attention for Stable and Interpretable Neural Networks

Introduction:
Attention mechanisms are a core component of modern artificial intelligence, especially in Transformer-based architectures. However, widely used formulations such as scaled dot-product attention are primarily heuristic and lack a clear energy-based interpretation. This limits theoretical understanding of their stability, robustness, and optimization behavior.

Methods:
In this work, we propose a novel attention mechanism inspired by the quantum harmonic oscillator (QHO). Query–key interactions are modeled as energy states within a bounded harmonic potential, where similarity scores are mapped to energy values using a Hamiltonian-based formulation. Instead of conventional softmax normalization, attention weights are computed through an exponential energy decay function motivated by quantum principles. We further analyze the proposed formulation to establish properties such as bounded gradients, Lipschitz continuity, and improved conditioning of the optimization landscape.

Results:
The proposed QHO-based attention is integrated into Transformer architectures and evaluated on standard classification and sequence modeling tasks. Experimental results show that it achieves performance comparable to conventional attention mechanisms while providing improved training stability and reduced sensitivity to initialization. Empirical analysis also indicates more controlled gradient behavior and enhanced robustness under noisy inputs and adversarial perturbations.

Conclusions:
This work introduces a physically interpretable and mathematically grounded alternative to traditional attention mechanisms. By framing attention through an energy-based perspective, it strengthens the theoretical foundation of neural architectures and offers a promising direction for building more stable, robust, and interpretable deep learning models.