Quantum mechanical framework for quantization-based optimization: from Gradient flow to Schrödinger equation

Jinwuk Seok, Changsik Cho

Keywords: Quantization, Optimization, Schrödinger equation, Fokker-Planck equation, Witten-Laplacian, Hamiltonian

Abstract:

This work presents a quantum mechanical framework for analyzing quantization-based optimization algorithms.
The sampling process of the quantization-based search is modeled as a gradient-flow dissipative system, leading to a Hamilton–Jacobi–Bellman (HJB) representation.
Through a suitable transformation of the objective function, this formulation yields the Schrödinger equation, which reveals that quantum tunneling enables escape from local minima and guarantees access to the global optimum.
By establishing the connection to the Fokker–Planck equation, the framework provides a thermodynamic interpretation of global convergence.
Such an analysis between the thermodynamic and quantum dynamic methodologies unifies combinatorial and continuous optimization, and extends naturally to machine learning tasks, such as image classification.
Numerical experiments demonstrate that quantization-based optimization consistently outperforms conventional algorithms across both combinatorial problems and nonconvex continuous functions.

Primary Area: optimization

Submission Number: 4222