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

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Akira SaiToh
Robabeh Rahimi
Mikio Nakahara

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First Online: 27 November 2013

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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

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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

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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

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