Yoshifumi Moriyama 1,a) Ichiro Iimura 2,b) Tomotsugu Ohno 1,c) Shigeru Nakayama 3,d)

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1 1,a) 2,b) 1,c) 3,d) Quantum-Inspired Evolutionary Algorithm 0-1 Search Performance Analysis According to Interpretation Methods for Dealing with Permutation on Integer-Type Gene-Coding Method based on Quantum Bit Representation Yoshifumi Moriyama 1,a) Ichiro Iimura 2,b) Tomotsugu Ohno 1,c) Shigeru Nakayama 3,d) Abstract: Quantum-inspired Evolutionary Algorithm (QEA) and QEA with Pair-Swap strategy (QEAPS), each gene is represented by a quantum bit (qubit) in both algorithms, have shown superior search performance than Classical Genetic Algorithm in 0-1 knapsack problem. And then, in the experimental result using integer knapsack problem, 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, have shown superior search performance than conventional binary-type gene-coding method. However, the gene-coding method cannot deal with permutation easily. Therefore, we have proposed two interpretation methods which can deal with permutation, in order to expand the gene-coding method based on the qubit representation. This paper has shown the results of computer experiment using the proposed interpretation methods, and has described the results of analysis. 1 National Institute of Technology, Ariake College, 150 Higashihagio-machi, Omuta, Fukuoka, Japan 2 Prefectural University of Kumamoto, Tsukide, Higashi-ku, Kumamoto, Japan 3 Emeritus Professor, Kagoshima University, Korimoto, Kagoshima, Japan a) yosifumi@ariake-nct.ac.jp b) iiimura@pu-kumamoto.ac.jp c) e47207@ga.ariake-nct.ac.jp d) shignaka@gmail.com 1. Quantum-Inspired Evolutionary Algorithm QEAClassical Genetic Algorithm: CGA[1] Knapsack Problem: 0-1KP [2], [3]QEA Pair Swap c 2015 Information Processing Society of Japan 1

2 Chromosome Chromosome Observed binary Observed binary 2 Fig. 2 Binary-type gene-coding method for solving binary optimization problems. Fig. 1 1 Gene-coding method based on qubit representation and the observed binary.! #"$& ('#"*),+('-.),/0 Chromosome Quantum-inspired Evolutionary Algorithm based on Pair SwapQEAPS0-1KP QEA [4], [5] Han Integer Knapsack Problem: IKP IKP 0-1KP [6], [7] Traveling Salesman Problem: TSP 2. QEA QEAPS 0-1KP IKP QEA QEAPS Fig. 3 Observed binary 3 Integer-type gene-coding method for solving integer optimization problems i q i q i = q i1 q i2 q im j = 1, 2,, m i j h j q ij q ij = q ij1 q ij2 q ijh k = 1, 2,, h q ijk 0 1 α ijk β ijk p ijk q ijk q ijk = α ijk 0 + β ijk 1 = [ αijk β ijk ]. (1) α ijk 2 0 β ijk 2 1 α ijk 2 + β ijk 2 = 1 c 2015 Information Processing Society of Japan 2

3 by unitary transform representation using integer-type gene-coding method Observe Rotation angle list Best solution Observed binary Create rotation angle list if and Decode Evaluate 4 IKP Fig. 4 cycle of an individual in IKP [2], [3] h = 1 j j p ij 1 p i q i f(p i ) p i CGA 2.3 [6], [7] j 3 j q ij h p ij = [p ij1, p ij2,, p ijh ] T x ij p ij x ij x i = [x i1, x i2,, x im ] q i f(x i ) x i CGA Table 1 1 IKP θ ijk Lookup table of the rotation angle θ ijk in IKP. θ ijk (for unitary transform) p ijk b ijk f(x i ) > f(s i ) α ij β ijk α ijk β ijk α ijk β ijk > 0 < 0 = 0 = false θ C θ C ±θ C 1 0 false θ C θ C ±θ C Otherwise IKP QEA QEAPS QEA QEAPS 4 h = 1 x ij x i = p i b i = s i i q i b i s i i q i p i p i x i x i p i x i [7] x i f(x i ) c 2015 Information Processing Society of Japan 3

4 by unitary transform representation using integer-type gene-coding method Observe Rotation angle list Create rotation angle list Best solution and if Permutation (Tour ) Evaluate Observed binary Decode Interpret as permutation 5 TSP Fig. 5 cycle of an individual in TSP. x i s i b i s i p i x i p i x i b i s i x i s i p i b i q i u i u i 1 [ α ijk β ijk ] = [ cos (θijk ) sin (θ ijk ) sin (θ ijk ) cos (θ ijk ) ] [ αijk β ijk ].(2) QEA 1 [2], [3]QEAPS 1 [4], [5] 3. TSP Table 2 2 TSP θ ijk Lookup table of the rotation angle θ ijk in TSP. θ ijk (for unitary transform) p ijk b ijk f(t i ) > f(s i ) α ij β ijk α ijk β ijk α ijk β ijk > 0 < 0 = 0 = false θ C θ C ±θ C 1 0 false θ C θ C ±θ C Otherwise TSP QEA QEAPS 5 TSP i q i b i s i i q i p i p i x i x i 5 x i1 x i3 6 m TSP x i x i t i t i f(t i ) t i s i b i s i p i t i t i s i 2 c 2015 Information Processing Society of Japan 4

5 Tour Tour (a) Example 1. (a) Example 1. Tour Tour (b) Example 2. (b) Example 2. 6 I 1 7 I 2 Fig. 6 Interpretation process using I 1 Fig. 7 Interpretation process using I 2 u i (2) q i IKP QEA QEAPS 3.2 TSP x i I 1 6 x i t i q i b i x i x i r i r ij j c rij c 1 A c 2 B c 3 C t i 6(a) 1 c ri1 c 5 E 2 c ri2 c 2 B 3 c ri3 c 4 D t i = E B D A C I 2 7 x i t i I 1 x i r i r ij c j t i c 2015 Information Processing Society of Japan 5

6 (0 + /. + * ()' " ' & $! # # &*-, <>= V2=?A@CB1DFEHG>I?W@/B1D EHG>I J K>LM*N O J KWX Y O <>= V2=?A@CB2G>IJ?W@/B2GUI K>L J KWX M N O Y O PQ= Z[=?R@-BSDFEHG>I?A@\BSDFE]G>I J K>LM*NFO J KWX Y O PQ= Z[=?R@-BTG>I?A@\B2G>I J KUL J KRX M*NFO Y O 8 (*)+-, +/. )102, ,:9*; I 1 Fig. 8 Experimental result in the case of I 1. CED ]:D FHG7IKJMLONEPQ5RTSU VW FOG^I9JMLONEPQ RZ_ ` W CED ]:D FHG7IXNEPQ FOG^IXNZPQ5RZ_ RTSU V W ` W Y a D FZG7IKJML[NZPQ5RTSU V W D FOGbIKJLcNEPQ5RZ_ ` W Y a D FZG7IXNHPQ5R\SU VW DFOGbIdNHPQ5RH_ ` W :48 4 A B I 2 Fig. 9 Experimental result in the case of I 2. 7(a) c 1 A 5 c 2 B 2 c 3 C 4, t i = D B E C A 4. TSPLIB *1 burma1414a B N C D E F L G M H K I J3,323 I 1 I 2 QEA QEAPS m = 14 h = ceil (log 2 14) = 4 QEAQEAPS 100 1,000,000 QEA 5 *1 TSPLIB95/ "! &# &# $ # "! 100 θ C [rad] θ C = { π/1000, π/900, π/800, π/700, π/600, π/500, π/400, π/300, π/200, π/100 } rad 20 8 I 1 9 I 2 20 R G QEA QEAPS I 2 I 1 I 1 6 x i3 6 2 t 2 t 3 t 5 BDC 6(a) CBD 6(b) I x i3 6 2 x i3 C 2 BE I 2 I 1 building block G R QEA θ C = { π/700, π/600 } rad QEAPS θ C = { π/300, π/200, π/100 } rad 5. TSP QEAPS TSP [1] D. E. Goldberg: Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, MA (1989). [2] K.-H. Han and J.-H. Kim: Quantum-inspired evolutionary algorithm for a class of combinatorial optimization, IEEE Trans. Evolutionary Computation, Vol. 6, No. 6, pp (2002). [3] K.-H. Han and J.-H. Kim: On setting the parameters of qea for practical applications: Some guidelines based on c 2015 Information Processing Society of Japan 6

7 empirical evidence, Genetic and Evolutionary Computation GECCO 2003, pp (2003). [4] S. Nakayama, T. Imabeppu, and S. Ono: Pair swap strategy in quantum-inspired evolutionary algorithm, Genetic and Evolutionary Computation GECCO 2006, Late Breaking Paper, Seattle, Washington, USA (2006). [5],,, :, D, Vol. J89-D, No. 9, pp (2006). [6] I. Iimura, Y. Moriyama, and S. Nakayama: Integer-type gene-coding method based on quantum bit representation in quantum-inspired evolutionary algorithm: Application to integer knapsack problem, J. Signal Processing, Vol. 16, No. 6, pp (2012). [7],, :,, Vol. 55, No. 2, pp (2014). c 2015 Information Processing Society of Japan 7