Quantum Information Processing

, Volume 13, Issue 3, pp 737–755 | Cite as

A quantum genetic algorithm with quantum crossover and mutation operations

Article

Abstract

In the context of evolutionary quantum computing in the literal meaning, a quantum crossover operation has not been introduced so far. Here, we introduce a novel quantum genetic algorithm that has a quantum crossover procedure performing crossovers among all chromosomes in parallel for each generation. A complexity analysis shows that a quadratic speedup is achieved over its classical counterpart in the dominant factor of the run time to handle each generation.

Keywords

Genetic algorithm Quantum computing Computational complexity 

Notes

Acknowledgments

A.S. is thankful to Shigeru Yamashita for his comment. A.S. and M.N. were supported by the “Open Research Center” Project for Private Universities: matching fund subsidy from MEXT. R.R. is supported by Industry Canada and CIFAR.

References

  1. 1.
    Ahuja, A., Kapoor, S.: A Quantum Algorithm for Finding the Maximum (1999). arXiv:quant-ph/9911082Google Scholar
  2. 2.
    Barnum, H., Bernstein, H.J., Spector, L.: A Quantum Circuit for OR (1999). arXiv:quant-ph/9907056Google Scholar
  3. 3.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschr. Phys. 46, 493–505 (1998)CrossRefGoogle Scholar
  4. 4.
    Brassard, G., Høyer, P., Tapp, A.: Quantum counting. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) Proceedings of Automata, Languages and Programming, 25th International Colloquium (ICALP’98) (LNCS 1443), pp. 820–831. Aalborg, Denmark, 13–17 July 1998, Springer, Berlin (1998). arXiv:quant-ph/9805082Google Scholar
  5. 5.
    Chakraborty, S., Radhakrishnan, J., Raghunathan, N.: Bounds for error reduction with few quantum queries. In: Chekuri, C., Jansen, K., Rolim, J., Trevisan, L. (eds.) Proceedings of the 9th International Workshop on Randomization and Computation (RANDOM 2005) (LNCS 3624), pp. 245–256. Berkeley, CA, 22–24 August 2005. Springer, Berlin (2005)Google Scholar
  6. 6.
    Chen, M., Quan, H.: Quantum-inspired evolutionary algorithm based on estimation of distribution. In: Proceedings of the 2nd International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA 2007), pp. 17–19. Zhengzhou, China, 14–17 September 2007. IEEE Press, Piscataway, NJ (2007)Google Scholar
  7. 7.
    Ding, S., Jin, Z., Yang, Q.: Evolving Quantum Oracles with Hybrid Quantum-Inspired Evolutionary Algorithm (2006). arXiv:quant-ph/0610105Google Scholar
  8. 8.
    Dürr, C., Høyer, P.: A Quantum Algorithm for Finding the Minimum (1996). arXiv:quant-ph/9607014Google Scholar
  9. 9.
    Gepp, A., Stocks, P.: A Review of Procedures to Evolve Quantum Algorithms (2007). arXiv:0708.3278Google Scholar
  10. 10.
    Giraldi, G.A., Portugal, R., Thess, R.N.: Genetic Algorithms and Quantum Computation (2004). arXiv:cs/0403003Google Scholar
  11. 11.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA (1989)MATHGoogle Scholar
  12. 12.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC 1996), pp. 212–219. Philadelphia, PA, 22–24 May 1996. ACM Press, New York, NY (1996)Google Scholar
  13. 13.
    Grover, L.K.: Quantum Search on Structured Problems (1998). arXiv:quant-ph/9802035Google Scholar
  14. 14.
    Grover, L.K.: Fixed-point quantum search. Phys. Rev. Lett. 95, 150501-1–150501-4 (2005)Google Scholar
  15. 15.
    Gruska, J.: Quantum Computing. McGraw-Hill, London (1999)Google Scholar
  16. 16.
    Han, K.H., Kim, J.H.: Genetic quantum algorithm and its application to combinatorial optimization problem. In: Proceedings of the 2000 Congress on Evolutionary Computation (CEC2000), pp. 1354–1360. La Jolla, CA, 16–19 July 2000. IEEE Press, Piscataway, NJ (2000)Google Scholar
  17. 17.
    Han, K.H., Kim, J.H.: Quantum-inspired evolutionary algorithm for a class of combinatorial optimization. IEEE Trans. Evol. Comput. 6(6), 580–593 (2002)CrossRefGoogle Scholar
  18. 18.
    Han, K.H., Kim, J.H.: Quantum-inspired evolutionary algorithms with a new termination criterion, \(h_{\epsilon }\) gate, and two-phase scheme. IEEE Trans. Evol. Comput. 8(2), 156–169 (2004)CrossRefGoogle Scholar
  19. 19.
    Holland, J.H.: Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence. The University of Michigan Press, Ann Arbor, MI (1975)Google Scholar
  20. 20.
    Johannsen, D., Kuru, P.P., Lengler, J.: Can quantum search accelerate evolutionary algorithms? In: Pelikan, M., Branke, J. (eds.) Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference (GECCO-2010), pp. 1433–1440. Portland, OR, 7–11 July 2010. ACM, New York, NY (2010)Google Scholar
  21. 21.
    Knuth, D.E.: The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed, Chap. 3. Addison-Wesley, Reading, MA (1997)Google Scholar
  22. 22.
    Leier, A., Banzhaf, W.: Evolving Hogg’s quantum algorithm using linear-tree GP. In: Cantú-Paz, E., Foster, J.A., Deb, K., Davis, L.D., Roy, R., O’Reilly, U.M., Beyer, H.G., Standish, R., Kendall, G., Wilson, S., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A.C., Dowsland, K.A., Jonoska, N., Miller, J. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference 2003 (GECCO-2003), Part I (LNCS 2723), pp. 390–400. Chicago, IL, 12–16 July 2003. Springer, Berlin (2003)Google Scholar
  23. 23.
    Leier, A., Banzhaf, W.: Comparison of selection strategies for evolutionary quantum circuit design. In: Deb, K. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference 2004 (GECCO-2004), Part II (LNCS 3103), pp. 557–568. Seattle, WA, 26–30 June 2004. Springer, Berlin (2004)Google Scholar
  24. 24.
    Liao, R., Wang, X., Qin, Z.: A novel quantum-inspired genetic algorithm with expanded solution space. In: Proceedings of the 2010 Second International Conference on Intelligent Human–Machine Systems and Cybernetics (IHMSC 2010), pp. 192–195. Nanjing, China, 26–28 August 2010. IEEE Computer Society, Los Alamitos, CA (2010)Google Scholar
  25. 25.
    Lukac, M., Perkowski, M.: Evolving quantum circuits using genetic algorithm. In: Stoica, A., Keymeulen, D., Lohn, J. (eds.) Proceedings of the 2002 NASA/DoD Conference on Evolvable Hardware, pp. 177–181. Alexandria, VA, 15–18 July 2002. IEEE Computer Society, Los Alamitos, CA (2002)Google Scholar
  26. 26.
    Lukac, M., Perkowski, M., Goi, H., Pivtoraiko, M., Yu, C.H., Chung, K., Jee, H., Kim, B.G., Kim, Y.D.: Evolutionary approach to quantum and reversible circuits synthesis. In: Yanushkevich, S.N. (ed.) Artificial Intelligence in Logic Design, pp. 201–257. Kluwer, Dordrecht (2004)CrossRefGoogle Scholar
  27. 27.
    Malossini, A., Blanzieri, E., Calarco, T.: QGA: quantum genetic algorithm (2004). Technical Report: #DIT-04-105, Dec. 2004, Univ. Trento, http://www.dit.unitn.it
  28. 28.
    Malossini, A., Blanzieri, E., Calarco, T.: Quantum genetic optimization. IEEE Trans. Evol. Comput. 12(2), 231–241 (2008)CrossRefGoogle Scholar
  29. 29.
    Massey, P., Clark, J.A., Stepney, S.: Evolving quantum circuits and programs through genetic programming. In: Deb, K. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference 2004 (GECCO-2004), Part II (LNCS 3103), pp. 569–580, Seattle, WA, 26–30 June 2004. Springer, Berlin (2004)Google Scholar
  30. 30.
    Massey, P., Clark, J.A., Stepney, S.: Human-competitive evolution of quantum computing artefacts by genetic programming. Evol. Comput. 14(1), 21–40 (2006)CrossRefGoogle Scholar
  31. 31.
    Matsumoto, M., Nishimura, T.: Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans. Model. Comput. Sim. 8, 3–30 (1998). http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/mt.html
  32. 32.
    Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge, MA (1996)Google Scholar
  33. 33.
    Mohammed, A.M., Elhefnawy, N.A., El-Sherbiny, M.M., Hadhoud, M.M.: Quantum crossover based quantum genetic algorithm for solving non-linear programming. In: Proceedings of the 8th International Conference on INFOrmatics and Systems (INFOS2012), pp. BIO-145-153. Cairo, Egypt, 14–16 May 2012. IEEE, Piscataway, NJ (2012)Google Scholar
  34. 34.
    Nakayama, S., Imabeppu, T., Ono, S.: Pair swap strategy in quantum-inspired evolutionary algorithm (2006). In: The Late-breaking papers of the 2006 Genetic and Evolutionary Computation Conference (GECCO-2006), Seattle, WA, 8–12 July 2006Google Scholar
  35. 35.
    Nakayama, S., Imabeppu, T., Ono, S., Iimura, I.: Consideration on pair swap strategy in quantum-inspired evolutionary algorithm. IEICE Trans. Inf. Sys. J89-D(9), 2134–2139 (2006) (in Japanese)Google Scholar
  36. 36.
    Narayanan, A., Moore, M.: Quantum-inspired genetic algorithms. In: Proceedings of the IEEE 3rd International Conference on Evolutionary Computation (ICEC96), pp. 61–66. Nagoya, Japan, 20–22 May 1996. IEEE Press, Piscataway, NJ (1996)Google Scholar
  37. 37.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  38. 38.
    Rubinstein, B.I.P.: Evolving quantum circuits using genetic programming. In: Proceedings of the 2001 Congress on Evolutionary Computation (CEC2001), pp. 144–151. Seoul, Korea, 27–30 May 2001. IEEE Press, Piscataway, NJ (2001)Google Scholar
  39. 39.
    Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E., Leigh, S., Levenson, M., Vangel, M., Banks, D., Heckert, A., Dray, J., Vo, S.: A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications (2010). NIST Special Publication 800-22, Revision 1a, http://csrc.nist.gov/groups/ST/toolkit/rng/index.html
  40. 40.
    Rylander, B., Soule, T., Foster, J., Alves-Foss, J.: Quantum evolutionary programming. In: Spector, L., Goodman, E.D., Wu, A., Langdon, W.B., Voigt, H.M., Gen, M., Sen, S., Dorigo, M., Pezeshk, S., Garzon, M.H., Burke, E. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), pp. 1005–1011. San Francisco, CA, 7–11 July 2001. Morgan Kaufmann, San Francisco (2001)Google Scholar
  41. 41.
    Sofge, D.A.: Prospective algorithms for quantum evolutionary computation. In: Bruza, P.D., Lawless, W., van Rijsbergen, K., Sofge, D.A., Coecke, B., Clark, S. (eds.) Proceedings of the 2nd Quantum Interaction Symposium (QI-2008), pp. 98–105. Oxford, UK, 26–28 March 2008. College Publications, London (2008). arXiv:0804.1133Google Scholar
  42. 42.
    Soklakov, A.N., Schack, R.: Efficient state preparation for a register of quantum bits. Phys. Rev. A 73, 012307-1–012307-13 (2006)Google Scholar
  43. 43.
    Spector, L.: Automatic Quantum Computer Programming: A Genetic Programming Approach. Springer, New York (2004, Paperback Ed. 2007)Google Scholar
  44. 44.
    Spector, L., Barnum, H., Bernstein, H.: Genetic programming for quantum computers. In: Koza, J.R. (eds.) Genetic Programming 1998: Proceedings of the Third Annual Conference (GP-98), pp. 365–374. Madison, WI, 22–25 July 1998. Morgan Kaufmann, San Francisco (1998)Google Scholar
  45. 45.
    Spector, L., Barnum, H., Bernstein, H., Swamy, N.: Finding a better-than-classical quantum AND/OR algorithm using genetic programming. In: Proceedings of the 1999 Congress on Evolutionary Computation (CEC1999), pp. 2239–2246. Washington, D.C., 6–9 July 1999. IEEE Press, Piscataway, NJ (1999)Google Scholar
  46. 46.
    Spector, L., Klein, J.: Machine invention of quantum computing circuits by means of genetic programming. AI EDAM 22, 275–283 (2008)Google Scholar
  47. 47.
    Tanaka, Y., Ichikawa, T., Tada-Umezaki, M., Ota, Y., Nakahara, M.: Quantum oracles in terms of universal gate set. Int. J. Quant. Inf. 9, 1363–1381 (2011)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Tulsi, T., Grover, L.K., Patel, A.: A new algorithm for fixed point quantum search. Quant. Inf. Comput. 6, 483–494 (2006)MathSciNetMATHGoogle Scholar
  49. 49.
    Udrescu, M., Prodan, L., Vlăduţiu, M.: Grover’s Algorithm and the Evolutionary Approach of Quantum Computation (2004). ACSA Report, “Politehnica” University of Timisoara, 15 Oct. 2004. http://www.acsa.upt.ro/publications/index.htm
  50. 50.
    Udrescu, M., Prodan, L., Vlăduţiu, M.: Implementing quantum genetic algorithms: a solution based on Grover’s algorithm. In: Proceedings of the 3rd Conference on Computing Frontiers, pp. 71–81. Ischia, Italy, 3–5 May 2006. ACM Press, New York (2006)Google Scholar
  51. 51.
    Ventura, D., Martinez, T.: Initializing the amplitude distribution of a quantum state. Found. Phys. Lett. 12, 547–559 (1999)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Williams, C.P., Gray, A.G.: Automated design of quantum circuits. In: Williams, C.P. (eds.) Quantum Computing and Quantum Communications: First NASA International Conference (LNCS 1509), pp. 113–125. Palm Springs, CA, 17–20 February 1998. Springer, Berlin (1999)Google Scholar
  53. 53.
    Yabuki, T., Iba, H.: Genetic algorithms for quantum circuit design-evolving a simpler teleportation circuit. In: Whitley, L.D., Goldberg, D.E., Cantú-Paz, E., Spector, L., Parmee, I.C., Beyer, H.G. (eds.) Proceedings of the 2000 Genetic and Evolutionary Computation Conference (GECCO-2000), pp. 425–430. Las Vegas, NV, 8–12 July 2000. Morgan Kaufmann, San Francisco (2000)Google Scholar
  54. 54.
    Zhang, G.: Quantum-inspired evolutionary algorithms: a survey and empirical study. J. Heuristics 17, 303–351 (2011)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Akira SaiToh
    • 1
    • 4
    • 5
  • Robabeh Rahimi
    • 2
  • Mikio Nakahara
    • 3
    • 4
  1. 1.Quantum Information Science Theory GroupNational Institute of InformaticsChiyodaJapan
  2. 2.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  3. 3.Department of PhysicsKinki UniversityHigashi-OsakaJapan
  4. 4.Research Center for Quantum Computing, Interdisciplinary Graduate School of Science and EngineeringKinki UniversityHigashi-OsakaJapan
  5. 5.Department of Computer Science and EngineeringToyohashi University of TechnologyTenpaku-choJapan

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