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

TL;DR

This paper introduces a quantum mechanical framework for quantization-based optimization that models the search process as a quantum system, enabling escape from local minima and improving performance in complex optimization tasks.

Contribution

It develops a novel quantum framework linking gradient flow, Schrödinger equation, and thermodynamics, unifying combinatorial and continuous optimization methods.

Findings

Quantum tunneling aids in escaping local minima.

The framework guarantees access to the global optimum.

Numerical results show superior performance over traditional algorithms.

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 Schroedinger 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 the quantum dynamic methodology 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.