A Formulation of Quantum Particle Swarm Optimization

Abstract

In this article, we present our version of quantum particle swarm optimization. The objective function is the expectation value of the quantum mechanical Hamiltonian, as the sum of the kinetic term ($-h^2 \frac{d^2}{dx^2}$) and the polynomial objective function ($V(x)$). The quantum mechanical feature in the system is represented by the set of expectation values $(\phi|x^iu^j|\phi)$ for the possible combination of the kinetic momentum operator ($u=h\frac{d}{dx}$) and the positional operator ($x$). These elements should satisfy the semi-definite condition of a matrix which is the representation of the uncertainty principle. The Plank constant $h$ is a control parameter which would switch from the quantum mechanical case to the classical case, and vice versa. In order to get the optimum, these quantum mechanical expectation values are assumed to be the particle coordinates and propagated by the particle swarm optimization with the restriction. We present a simple example of the computational process.