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The quantum-inspired evolutionary algorithm (QEA) is one of the evolutionary algorithms incorporating principles of quantum computation. In QEA, each gene is represented by a quantum bit (qubit), and the quantum superposition state is imitated. QEA can effectively shift from a global search to a local search. Han, et al. showed that QEA has superior search performance to a genetic algorithm (GA) in the 0-1 knapsack problem (0-1KP). Nakayama, et al. proposed a simpler algorithm that is referred to as QEA based on pair swap (QEAPS). QEAPS requires fewer parameters to be adjusted than QEA. Nakayama, et al. showed that QEAPS can find similar or even better quality solutions than QEA in 0-1KP. However, in QEA and QEAPS, each gene is represented by a qubit and both algorithms can only use a binary value as an observation result for a qubit. Therefore, Iimura, et al. proposed a novel integer-type gene-coding method that can obtain an integer value as an observation result by assigning multiple qubits in a gene locus. Moreover, they implemented the gene-coding method in both QEA and QEAPS and showed that it can search for similar or even better quality solutions in a shorter time than a conventional binary-type gene-coding method in the integer knapsack problem (IKP). However, the integer-type gene-coding method cannot deal with permutations simply. In order to expand the gene-coding method based on the qubit representation, Moriyama, et al. proposed two interpretation methods based on the integer-type gene-coding method that can deal with permutations. Also, they clarified that the two proposed interpretation methods can search for the optimal solution, even with the genecoding method based on the qubit representation. This paper proposes a new gene-coding method that can deal with permutations. The proposed method promises effect of improving a solution in permutation space, like the k-Opt method, and can search for the optimal solution of the traveling salesman problem (TSP) effectively. From experimental results for TSP, the discovery rate of the optimal solution using the proposed method is higher than using conventional method in many cases of QEA and QEAPS.

Permutation Optimization Using a Gene-Coding Method of Quantum Bit by Local Improvement like the k-Opt