Quantum Dynamics of Machine Learning
Peng Wang, Maimaitiniyazi Maimaitiabudula
TL;DR
This work addresses the lack of a rigorous dynamical theory for iterative machine learning by casting the optimization in a quantum-dynamics framework via $i\frac{\partial \psi}{\partial t}= [-D \frac{\partial^{2}}{\partial x^{2}} + \zeta(x)]\psi$, with $\zeta(x)$ the generalized objective. By performing a Wick rotation ($\tau=it$), it links quantum dynamics to classical diffusion and thermodynamics, enabling a two-loop learning interpretation: quantum annealing through decreasing $D$ and evolution toward the ground state at fixed $D$. The paper provides (i) a structured derivation of ML dynamics from the Schrödinger equation, (ii) a Taylor-series-based approximation of $\zeta(x)$, (iii) a classical diffusion-reaction view, (iv) a derivation of Softmax and Sigmoid from energy-based sampling, and (v) a diffusion-model interpretation within this quantum framework. These contributions offer a rigorous mathematical foundation for ML iterations and point toward potential quantum-accelerated ML implementations.
Abstract
The quantum dynamic equation (QDE) of machine learning is obtained based on Schrödinger equation and potential energy equivalence relationship. Through Wick rotation, the relationship between quantum dynamics and thermodynamics is also established in this paper. This equation reformulates the iterative process of machine learning into a time-dependent partial differential equation with a clear mathematical structure, offering a theoretical framework for investigating machine learning iterations through quantum and mathematical theories. Within this framework, the fundamental iterative process, the diffusion model, and the Softmax and Sigmoid functions are examined, validating the proposed quantum dynamics equations. This approach not only presents a rigorous theoretical foundation for machine learning but also holds promise for supporting the implementation of machine learning algorithms on quantum computers.