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.

翻译：本文提出了一种用于分析基于量子化的优化算法的量子力学框架。该框架将量子化搜索的采样过程建模为一个梯度流耗散系统，从而导出了 Hamilton-Jacobi-Bellman (HJB) 方程表示。通过对目标函数进行适当的变换，该表述进一步推导出薛定谔方程，揭示了量子隧穿效应能够使算法逃离局部极小值并保证获得全局最优解。通过建立与 Fokker-Planck 方程的联系，该框架为全局收敛性提供了热力学解释。这种热力学与量子动力学方法论之间的分析，统一了组合优化与连续优化，并可自然地推广至如图像分类等机器学习任务。数值实验表明，基于量子化的优化算法在组合问题与非凸连续函数优化上均持续优于传统算法。