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Open Journal of Optimization

Quantum Inspired Differential Evolution Algorithm

Binxu Li, Panchi Li

Published Jan 1, 2015• 11 citations
Publisher

Abstract

To enhance the optimization performance of differential evolution algorithm, by studying the implementation mechanism of differential evolution algorithm, a new idea of incorporating differential strategy and rotation of qubits in the Bloch sphere is proposed in this paper. In the proposed approach, the individuals are encoded by qubits described on Bloch sphere, and the rotation angles of qubits in current individual are obtained by differential strategy. The axis of rotation is designed by using vector product theory, and the rotation matrixes are constructed by using Pauli matrixes. Taking the corresponding qubits in current best individual as targets, the qubits in current individual are rotated to the target qubits about the rotation axis on the Bloch sphere. The Hadamard gates are used to mutate individuals. The simulation results of optimizing the minimum value of functions indicate that, for an iterative step, the average time of the proposed approach is 13 times as long as that of the classical differential evolution algorithm. When the same limited steps are applied in two approaches, the average optimization result of the proposed approach is 0.3 times as great as that of the classical differential evolution algorithm; when the same running time is applied in two approaches, the average optimization result of the proposed approach is 0.4 times as great as that of the classical differential evolution algorithm. These results suggest that the proposed approach is inefficient in computational ability; however, it is obviously efficient in optimization ability, and the overall optimization performance is better than that of the classical differential evolution algorithm.

DOI
10.4236/ojop.2015.42004

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To enhance the optimization performance of differential evolution algorithm, by studying the implementation mechanism of differential evolution algorithm, a new idea of incorporating differential strategy and rotation of qubits in the Bloch sphere is proposed in this paper. In the proposed approach, the individuals are encoded by qubits described on Bloch sphere, and the rotation angles of qubits in current individual are obtained by differential strategy. The axis of rotation is designed by using vector product theory, and the rotation matrixes are constructed by using Pauli matrixes. Taking the corresponding qubits in current best individual as targets, the qubits in current individual are rotated to the target qubits about the rotation axis on the Bloch sphere. The Hadamard gates are used to mutate individuals. The simulation results of optimizing the minimum value of functions indicate that, for an iterative step, the average time of the proposed approach is 13 times as long as that of the classical differential evolution algorithm. When the same limited steps are applied in two approaches, the average optimization result of the proposed approach is 0.3 times as great as that of the classical differential evolution algorithm; when the same running time is applied in two approaches, the average optimization result of the proposed approach is 0.4 times as great as that of the classical differential evolution algorithm. These results suggest that the proposed approach is inefficient in computational ability; however, it is obviously efficient in optimization ability, and the overall optimization performance is better than that of the classical differential evolution algorithm.
To enhance the optimization performance of differential evolution algorithm,by studying the implementation mechanism of differential evolution algorithm,a new idea of incorporating differential strategy and rotation of qubits in the Bloch sphere is proposed in this paper.In the proposed approach,the individuals are encoded by qubits described on Bloch sphere,and the rotation angles of qubits in current individual are obtained by differential strategy.The axis of rotation is designed by using vector product theory,and the rotation matrixes are constructed by using Pauli matrixes.Taking the corresponding qubits in current best individual as targets,the qubits in current individual are rotated to the target qubits about the rotation axis on the Bloch sphere.The Hadamard gates are used to mutate individuals.The simulation results of optimizing the minimum value of functions indicate that,for an iterative step,the average time of the proposed approach is 13times as long as that of the classical differential evolution algorithm.When the same limited steps are applied in two approaches,the average optimization result of the proposed approach is 0.3 times as great as that of the classical differential evolution algorithm; when the same running time is applied in two approaches,the average optimization result of the proposed approach is 0.4 times as great as that of the classical differential evolution algorithm.These results suggest that the proposed approach is inefficient in computational ability; however,it is obviously efficient in optimization ability,and the overall optimization performance is better than the classical differential evolution algorithm.
To enhance the performance of the intelligent optimization algorithm, a new model of performing search on the Bloch sphere is proposed. Then, by integrating the model into the artificial fish swarm optimization, we present a quantum-inspired artificial fish swarm optimization algorithm. In proposed method, the fishes are encoded with the qubits described on the Bloch sphere. The vector product theory is adopted to establish the axis of rotation, and the Pauli matrices are used to construct the rotation matrices. The four fish behaviors, such as moving, tracking, capturing, aggregating, are achieved by rotating the current qubit about the rotation axis to the target qubit on the Bloch sphere. The Bloch coordinates of qubit can be obtained by measuring with the Pauli matrices, and the optimization solutions can be presented through the solution space transformation. The highlight advantages of this method are the ability to simultaneously adjust two parameters of a qubit and automatically achieve the best match between two adjustment quantities, which may accelerate the optimization process. The experimental results show that the proposed method obviously outperforms the classical one in convergence speed and achieves better levels for some benchmark functions.
To enhance the performance of the artificial bee colony optimization by integrating the quantum computing model into bee colony optimization, we present a quantum-inspired bee colony optimization algorithm. In our method, the bees are encoded with the qubits described on the Bloch sphere. The classical bee colony algorithm is used to compute the rotation axes and rotation angles. The Pauli matrices are used to construct the rotation matrices. The evolutionary search is achieved by rotating the qubit about the rotation axis to the target qubit on the Bloch sphere. By measuring with the Pauli matrices, the Bloch coordinates of qubit can be obtained, and the optimization solutions can be presented through the solution space transformation. The proposed method can simultaneously adjust two parameters of a qubit and automatically achieve the best match between two adjustment quantities, which may accelerate the optimization process. The experimental results show that the proposed method is obviously superior to the classical one for some benchmark functions.
In the existing quantum intelligent optimization algorithms,almost all of the individuals are encoded by qubits described on plane unit circle.Since there is only one adjustable parameter,quantum properties have not been fully embodied,which limits the improvement of the optimization ability further.To address this issue,a chaos quantum immune algorithm based on Bloch sphere is proposed.In the proposed algorithm,the antibodies are encoded by qubits described on Bloch sphere,the axis of rotation is designed by using Pauli matrix,the clone of the excellent antibodies is performed by rotating qubits on Bloch sphere,and the local searching is achieved by employing the chaos variables in the rotation angles.The global searching is achieved by the mutations of the inferior individuals based on the Hadamard gates.The simulation results show that the proposed algorithm is superior to other quantum intelligent optimization algorithms in both search capability and optimization efficiency.
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.
The current quantum evolution algorithms have slow convergence rate and poor robustness. In order to overcome the two shortages, a novel self-adaptive quantum genetic algorithm is proposed. Firstly, the new algorithm adopts an encoding method which is based on the Bloch spherical coordinates. Secondly, in the process of searching the optimal solution, a self-adaptive factor is introduced to reflect the relative change rates which are relative to the difference of the best individual's objective fitness between the parent generation and the child generation. The convergence rate and direction of the algorithm can be improved by adjusting the factor. The rules of updating the rotation angle and are constructed. Finally, using hadamard gate of the quantum in the mutation strategy, it can enhance the diversity of population. The simulation results of the optimizing problem of the multidimensional complex functions show that the new algorithm has not only avoided effectively the premature and improved the convergence rate, but also boosted strikingly efficiency and stability robustness of the algorithm.