A quantum genetic algorithm with quantum crossover and mutation operations

Akira SaiToh; Robabeh Rahimi; Mikio Nakahara

doi:10.1007/s11128-013-0686-6

Quantum Information Processing

March 2014, Volume 13, Issue 3, pp 737–755 | Cite as

A quantum genetic algorithm with quantum crossover and mutation operations

Authors
Authors and affiliations

Akira SaiToh
Robabeh Rahimi
Mikio Nakahara

Article

First Online: 27 November 2013

Received: 19 November 2012

Accepted: 06 November 2013

457 Downloads
9 Citations

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

This is a preview of subscription content, to check access.

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.

Appendix: An example of constructing the pseudo-randomizer \(R\)

In this appendix, we show an example to construct the pseudo-randomizer \(R\) used in Algorithms 2, 3, and 4. It should map a \(c\)-bit string to a pseudo-random \(n\)-bit string except for \(0_0\cdots 0_{c-1}\) that is mapped to \(0_0\cdots 0_{n-1}\). Its internal circuit complexity should be \(\mathrm{poly}(cn)\). As we use a quantum circuit to realize it in Algorithm 2, it is desirable to employ a circuit structure that is originally unitary.

Consider inputs \(a\in \{0,1\}^c\). We design a circuit that maps \(a_0\cdots a_{c-1}0_0\cdots 0_{n-1}\) to \(a_0\cdots a_{c-1}\kappa _0\cdots \kappa _{n-1}\) with \(\kappa =R(a)\), an \(n\)-bit pseudo-random number (here, \(a_0\cdots a_{c-1}\) and \(\kappa _0\cdots \kappa _{n-1}\) are the binary representations of \(a\) and \(\kappa \)). By the definition of \(R\), the circuit preserves \(0_0\cdots 0_{c-1}0_0\cdots 0_{n-1}\). This circuit is generated by function gen_r_circ() described in Fig. 9. As is clear from the description, the circuit output from this function can be directly used as a quantum circuit. Using the circuit \(C=\) gen_r_circ(), we have \(|a\rangle |0\rangle \overset{C}{\mapsto }|a\rangle |R(a)\rangle \) \(\forall a\in \{0,1\}^c\). The circuit complexity of \(C\) is \(O(cn)\) because, for each \(i\), at most \(n\) CNOT gates are used to decompose the gate output from step (2). In addition, gen_r_circ() spends \(O(c\;\mathrm{poly}(n))\) time when a common random number generator [21, 31] is used in step (1).

Fig. 9
Description of function gen_r_circ()

Note that the function gen_r_circ() is called only once for each \(t\), in the beginning of step 1 in Algorithms 2, 3, and 4. We have only to reuse the circuit \(C\) for the use of the pseudo-randomizer until \(t\) is incremented.

It is expected that outputs from the circuit \(C\) possess good uniformity if we use a good random number generator in step (1) of gen_r_circ() for generating \(C\). Let us write \(\gamma \) as \(\gamma (i)\) to emphasize its dependence on \(i\). For a nonzero input \(a_0\cdots a_{c-1}\), the \(k\)th bit of the output \(R(a)\) is \(\sum _{i=0}^{c-1}a_i\cdot \gamma _k(i)~~\mathrm{mod}~2\). This indicates that, for two different inputs \(a\) and \(a'\), the \(k\)th bits of \(R(a)\) and \(R(a')\) differ with probability \(1/2\) in the ideal case where \(\gamma (i)\)’s are generated from a true random number generator. This is because \(a\) and \(a'\) differ by at least a single bit. It also indicates that two different bits, the \(k\)th and the \(k'\)th bits, of \(R(a)\) for a nonzero input \(a\) differ with the probability \(1/2\) in the ideal case. This is because \(\gamma _k(i)\) and \(\gamma _{k'}(i)\) differ with the probability \(1/2\).

Now we show the result of our numerical test of \(C\). We tried statistical tests of randomness [21, 39] to test pseudo-random numbers output from \(C\), using NIST’s Statistical Test Suite (STS) (version 2.1.1) [39]. We set \(c=10\) and \(n=32\). Mersenne Twister (MT) (version mt19937ar) [31] was used to generate \(\gamma \) in step (1) of gen_r_circ(). We used the seed value 121,212 and did not reset MT during the circuit generation. The circuit \(C\) output from gen_r_circ(), of course, consisted of \(10\) outputs from step (2). For this \(C\), we used the inputs \(a\in \{0,1\}^c\backslash \{0_0\cdots 0_{c-1}\}\) from smaller to larger and obtained corresponding outputs \(R(a)\) by numerical computation. We obtained \(\hbox {1,023}\times 32\) bits in total in the outputs, since \(2^{10}-1=1023\). We regarded them as a serial bit string from left to right and used STS in its default setting to test the string. In the execution of STS, we used \(25\) binary sequences with length 1,200 as samples from the string. The following tests were tried with the default parameter values in STS: the Frequency Test, the Block Frequency Test, the Cumulative Sums Test, the Runs Test, the Longest-Run-of-Ones Test, the Binary Matrix Rank Test, the Spectral DFT Test, and the Serial Test. The string passed the tests except for the Binary Matrix Rank Test. It should be noted that the input length was too small for the binary matrix rank test [39]. In addition, randomness is not very strictly required for the use in evolutionary computing. Therefore, considering the tests that the string passed, we may claim that gen_r_circ() generates a usable pseudo-randomizer circuit for our algorithm.

We conducted another test: We generated ten circuits by calling gen_r_circ() ten times without resetting MT, using the seed value 676,767. For each circuit, we performed the same process as above to obtain the serial bit string. We obtained ten serial bit strings in total and tested the concatenated string using STS. As samples input to STS, we used \(25\) binary sequences with length 12,000. The concatenated string passed the tests except for the Binary Matrix Rank Test and the Spectral DFT Test. It was unexpected that it did not pass the spectral DFT test. It requires a further investigation to reveal the reason of this phenomenon.

The results of the first and the second tests are summarized in Table 1. In summary for this appendix, we found that a pseudo-randomizer circuit whose outputs possess enough randomness for the use in evolutionary computing can be generated by the function gen_r_circ(). It is hoped that the function will be improved so as to achieve better randomness for the sake of general use.

Table 1

List of test results for the first and the second tests we performed (see the text for the details of the tests)

	1st Test	2nd Test
Frequency Test	0.021262	0.186566
Block Frequency Test	0.105618	0.001156
Cumulative Sums Test (Forward)	0.105618	0.029796
Cumulative Sums Test (Reverse)	0.001691	0.057146
Runs Test	0.141256	0.010606
Longest-Run-of-Ones Test	0.875539	0.875539
Binary Matrix Rank Test	0.000000*	0.000000*
Spectral DFT Test	0.000533	0.000000*
Serial Test	0.041438	0.186566

Each value is a \(P\)-value (see Ref. [39] for its meaning for each test). * A failure to pass the test. We should mention that the values for the Longest-Run-of-Ones Test accidentally coincided, while the internally used results for sample sequences were different

References

1.
Ahuja, A., Kapoor, S.: A Quantum Algorithm for Finding the Maximum (1999). arXiv:quant-ph/9911082Google Scholar
2.
Barnum, H., Bernstein, H.J., Spector, L.: A Quantum Circuit for OR (1999). arXiv:quant-ph/9907056Google Scholar
3.
Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschr. Phys. 46, 493–505 (1998)CrossRefGoogle Scholar
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.
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.
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.
Ding, S., Jin, Z., Yang, Q.: Evolving Quantum Oracles with Hybrid Quantum-Inspired Evolutionary Algorithm (2006). arXiv:quant-ph/0610105Google Scholar
8.
Dürr, C., Høyer, P.: A Quantum Algorithm for Finding the Minimum (1996). arXiv:quant-ph/9607014Google Scholar
9.
Gepp, A., Stocks, P.: A Review of Procedures to Evolve Quantum Algorithms (2007). arXiv:0708.3278Google Scholar
10.
Giraldi, G.A., Portugal, R., Thess, R.N.: Genetic Algorithms and Quantum Computation (2004). arXiv:cs/0403003Google Scholar
11.
Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA (1989)zbMATHGoogle Scholar
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.
Grover, L.K.: Quantum Search on Structured Problems (1998). arXiv:quant-ph/9802035Google Scholar
14.
Grover, L.K.: Fixed-point quantum search. Phys. Rev. Lett. 95, 150501-1–150501-4 (2005)Google Scholar
15.
Gruska, J.: Quantum Computing. McGraw-Hill, London (1999)Google Scholar
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.
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.
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.
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.
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.
Knuth, D.E.: The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed, Chap. 3. Addison-Wesley, Reading, MA (1997)Google Scholar
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.
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.
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.
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.
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.
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.
Malossini, A., Blanzieri, E., Calarco, T.: Quantum genetic optimization. IEEE Trans. Evol. Comput. 12(2), 231–241 (2008)CrossRefGoogle Scholar
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.
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.
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.
Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge, MA (1996)Google Scholar
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.
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.
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.
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.
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
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.
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.
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.
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.
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.
Spector, L.: Automatic Quantum Computer Programming: A Genetic Programming Approach. Springer, New York (2004, Paperback Ed. 2007)Google Scholar
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.
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.
Spector, L., Klein, J.: Machine invention of quantum computing circuits by means of genetic programming. AI EDAM 22, 275–283 (2008)Google Scholar
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)MathSciNetCrossRefzbMATHGoogle Scholar
48.
Tulsi, T., Grover, L.K., Patel, A.: A new algorithm for fixed point quantum search. Quant. Inf. Comput. 6, 483–494 (2006)MathSciNetzbMATHGoogle Scholar
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.
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.
Ventura, D., Martinez, T.: Initializing the amplitude distribution of a quantum state. Found. Phys. Lett. 12, 547–559 (1999)MathSciNetCrossRefGoogle Scholar
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.
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.
Zhang, G.: Quantum-inspired evolutionary algorithms: a survey and empirical study. J. Heuristics 17, 303–351 (2011)CrossRefzbMATHGoogle Scholar

Copyright information

Authors and Affiliations

Akira SaiToh
- 1
- 4
- 5
Email author
Robabeh Rahimi
- 2
Mikio Nakahara
- 3
- 4

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

A quantum genetic algorithm with quantum crossover and mutation operations

Abstract

Keywords

Notes

Acknowledgments

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article