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Microcomputer applications

A Quantum Particle Swarm Optimization Algorithm for Solving Nonlinear Equations

Juan Du

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Aiming at the problems of the classical algorithms for solving the nonlinear equations such as high sensitivity to the initial value、poor convergence,a hybrid quantum genetic algorithm for solving nonlinear equations is proposed.The positions of particles are encoded by the probability amplitudes of quantum bits.The movements and mutations of particles are performed by quantum rotation gate and quantum non-gate.The algorithm has the abilities of population search for quantum swarm particle and global convergence.The quasi-newton method is combined into the algorithm and increases local searching ability and the high accuracy.Numerical simulation experiments show that this algorithm has reliable convergence,fast convergence rate and high precision.

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Aiming at the problems of the classical algorithms for solving the nonlinear equations such as high sensitivity to the initial value、poor convergence,a hybrid quantum genetic algorithm for solving nonlinear equations is proposed.The positions of particles are encoded by the probability amplitudes of quantum bits.The movements and mutations of particles are performed by quantum rotation gate and quantum non-gate.The algorithm has the abilities of population search for quantum swarm particle and global convergence.The quasi-newton method is combined into the algorithm and increases local searching ability and the high accuracy.Numerical simulation experiments show that this algorithm has reliable convergence,fast convergence rate and high precision.
To improve search ability and optimization efficiency and to avoid premature con- vergence for particle swarms optimization,a novel quantum particle swarm optimization for continuous space optimization is proposed.The positions of particles are encoded by the prob- ability amplitudes of quantum bits,the movements of particles are performed by quantum rotation gates,which achieve particles searching.The mutations of particles are performed by quantum non-gate,which increase the diversity Of particles.As each quantum bit contains two probability amplitudes,each particle occupies two positions in space.Hence,the process of searching is accelerated.The experimental results show that the algorithm proposed is superior to the basic particle swarms optimization.
In view of the shortcomings of particle swarm optimization such as premature convergence to local optimization, particle swarm optimization algorithm based on quantum gate and particle swarm optimization algorithm based on quantum behavior are studied in this paper. The first algorithm uses the random observation method of quantum bit coding particles to simulate quantum particle collapse for generating a population and uses the quantum rotating gate to generate a new population. The adaptive mutation operator is used for ensuring the diversity of the population, effectively reducing the impact of local optimization; therefore, the robustness of the algorithm is improved. The second algorithm uses the probability density function of quantum computation to make the particles jump out of the local extreme points and fulfils the global search, which is more suitable for continuous optimization. The results of computational experiments show that both the two particle swarm optimization algorithms based on quantum technology have a good global convergence.
This paper proposes a novel quantum genetic algorithm(QGA)——PSQGA,which uses particle swarm optimization method instead of quantum gate to update the state of quantum bit.It has the advantages of particle swarm optimization and quantum genetic algorithm.A variance function is introduced to estimate the stability of the algorithm.Though the experiments of four continuous functions and combination optimization problems,as well as its application to image sparse decomposition.Compared with the improved algorithm which involved quantum gate(IQGA),the ability of finding the best solution and the stability of PSQGA are greatly improved.PSQGA has better convergent property and ability of searching more extensive space.It is fit for the solution of complex optimization problems.
To improve the efficiency of Particle Swarm Optimization(PSO),a quantum particle swarm optimization algorithm combined with quantum theory on the basis of Bloch sphere was proposed.In Bloch spherical coordinates,the particle automatically updated rotation angle and particle position,without setting the rotation angle in the form of look-up table(or setting fixed value of the interval),making up for the deficiency of quantum evolutionary algorithm and quantum genetic algorithm on the basis of Bloch sphere,and the algorithm is more generalizable.Using quantum Hadamard gate to realize the variation of particle enhanced the diversity of population,and prompted particle jump out of local extreme value.The simulation results of the typical function optimization problem show that the algorithm is stable with high precision and fast convergence rate,and it is practical.