156 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 8, NO. 2, APRIL 2004 Quantum-Inspired Evolutionary Algorithms With a New Termination Criterion, Gate, and Two-Phase Scheme Kuk-Hyun Han, Associate Member, IEEE, and Jong-Hwan Kim, Senior Member, IEEE Abstract—From recent research on combinatorial optimization of the knapsack problem, quantum-inspired evolutionary algo- rithm (QEA) was proved to be better than conventional genetic algorithms. To improve the performance of the QEA, this paper proposes research issues on QEA such as a termination criterion, a -gate, and a two-phase scheme, for a class of numerical and combinatorial optimization problems. A new termination criterion is proposed which gives a clearer meaning on the convergence of -bit individuals. A novel variation operator gate, which is a modified version of the rotation gate, is proposed along with a two-phase QEA scheme based on the analysis of the effect of changing the initial conditions of -bits of the -bit individual in the first phase. To demonstrate the effectiveness and applicability of the updated QEA, several experiments are carried out on a class of numerical and combinatorial optimization problems. The results show that the updated QEA makes QEA more powerful than the previous QEA in terms of convergence speed, fitness, and robustness. Index Terms—Initial condition, numerical and combinatorial optimization, -bit representation, -gate, quantum-inspired evolutionary algorithm (QEA), termination criterion. I. INTRODUCTION E VOLUTIONARY algorithms (EAs) are principally a sto- chastic search and optimization method based on the prin- ciples of natural biological evolution. Compared with traditional optimization methods, such as calculus-based methods and enu- merative strategies, EAs are robust, global in operation, and may be applied generally without recourse to domain-specific heuristics, although their performance may be affected by these heuristics. Overviews of current state of the art in the field of evolutionary computation are given by Fogel [1] and Bäck [2]. EAs are characterized by the representation of the individual, the evaluation function representing the fitness level of the in- dividuals, and the population dynamics such as population size, variation operators, parent selection, reproduction and inheri- Manuscript received May 17, 2003; revised October 27, 2003. This work was supported in part by the Brain Korea 21 Project, School of Information Tech- nology, Korea Advanced Institute of Science and Technology (KAIST), and in part by the ITRC Intelligent Robot Research Center (IRRC) at KAIST supported by the Korea Ministry of Information and Communication in 2003. K.-H. Han was with the Department of Electrical Engineering and Com- puter Science, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Korea. He is now with the Digital Media R&D Center, Sam- sung Electronics Company, Ltd., Suwon, Gyeonggi 443-742, Korea (e-mail: khhan@ khhan.com). J.-H. Kim is with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Korea (e-mail: johkim@rit.kaist.ac.kr). Digital Object Identifier 10.1109/TEVC.2004.823467 Fig. 1. Quantum-inspired evolutionary algorithm (QEA). tance, survival competition method, etc. To have a good balance between exploration and exploitation, these components should be designed properly. The quantum-inspired evolutionary algorithm (QEA) re- cently proposed in [3] can treat the balance between exploration and exploitation more easily when compared with conventional genetic algorithms (CGAs). Also, QEA can explore the search space with a smaller number of individuals and exploit the search space for a global solution within a short span of time. QEA is based on the concept and principles of quantum computing, such as the quantum bit and the superposition of states. However, QEA is not a quantum algorithm, but a novel EA [4] as shown in Fig. 1. Like any other EAs, QEA is also characterized by the representation of the individual, the evaluation function, and the population dynamics. Quantum computing is a research area that includes concepts like quantum mechanical computers and quantum algorithms. Quantum mechanical computers were proposed in the early 1980s [5], [6] and their description was formalized in the late 1980s [7], [8]. Many efforts on quantum computers have pro- gressed actively since the early 1990s because these computers were shown to be more powerful than digital computers for solving various specialized problems. There are well-known quantum algorithms such as Deutsch-Jozsa algorithm [9], Simon’s algorithm [10], Shor’s quantum factoring algorithm [11], [12], and Grover’s database search algorithm [13], [14]. In particular, since the difficulty of the factoring problem is crucial for the security of the RSA cryptosystem [15] which is in widespread use today, interest in quantum computing is increasing [16]. Research on merging evolutionary computation and quantum computing began in the late 1990s. It can be classified into two 1089-778X/04$20.00 © 2004 IEEE