Quantum Dynamics of Optimization Problems

TL;DR

This paper introduces a quantum framework for optimization problems by formulating them as Schr"odinger equations, linking the objective function to wave functions, and deriving Fokker-Planck equations to analyze algorithm dynamics.

Contribution

It establishes a novel quantum interpretation of optimization problems and derives dynamic equations to study algorithm behavior theoretically.

Findings

Quantum interpretation of optimization problems established

Fokker-Planck equation derived for algorithm dynamics

Provides a theoretical basis for analyzing optimization algorithms

Abstract

In this letter, by establishing the Schr\"odinger equation of the optimization problem, the optimization problem is transformed into a constrained state quantum problem with the objective function as the potential energy. The mathematical relationship between the objective function and the wave function is established, and the quantum interpretation of the optimization problem is realized. Under the black box model, the Schr\"odinger equation of the optimization problem is used to establish the kinetic equation, i.e., the Fokker-Planck equation of the time evolution of the optimization algorithm, and the basic iterative structure of the optimization algorithm is given according to the interpretation of the Fokker-Planck equation. The establishment of the Fokker-Planck equation allows optimization algorithms to be studied using dynamic methods and is expected to become an important theoretical basis for algorithm dynamics.