*3.6. Crossover Operation*

The crossover operation is another main operation in differential evolution. The trial vector *qCt θi* is generated by crossover between the mutant vector *q*M*<sup>t</sup> θi* and the target vector *qtθi*with a binomial crossover strategy [44].

The quantum trial vector *q*C*<sup>t</sup> θi*at the *t*th generation can be generated by:

$$q\_{\theta ij}^{\mathsf{C}t} = \begin{cases} \ q\_{\theta ij'}^{\mathsf{Mt}} \text{ if } \left( rand\_j(0, 1) \le CR^t \right) \text{ or } (j = rnbr\_-i) \\\ q\_{\theta ij'}^{t} \text{ if } \left( rand\_j(0, 1) > CR^t \right) \text{ and } (j \ne rnbr\_-i) \end{cases} \tag{12}$$

where *CR* ∈ [0, 1] is the probability of the crossover operation which is randomly generated at *t*th iteration. In addition, *rnbr*\_*i* ∈ {1, *m*} is an integer to ensure that *q*C*<sup>t</sup> θi* obtains at least one vector from *q*M*<sup>t</sup> θi*.

 The quantum trial individual *q*C*<sup>t</sup> i* corresponding to the quantum trial vector *q*C*<sup>t</sup> θi* can be obtained by:

$$q\_i^{\mathbf{C}t} = \begin{bmatrix} \cos \theta\_{i1}^{\mathbf{C}t} & \cos \theta\_{i2}^{\mathbf{C}t} & \dots & \cos \theta\_{ill}^{\mathbf{C}t} \\ \sin \theta\_{i1}^{\mathbf{C}t} & \sin \theta\_{i2}^{\mathbf{C}t} & \dots & \sin \theta\_{im}^{\mathbf{C}t} \end{bmatrix} \tag{13}$$

where *m* is the length of the qubit quantum individual.

The quantum trial population *Q*<sup>C</sup>(*t*) = *q*C*<sup>t</sup>* 1 , *q*C*<sup>t</sup>* 2 ,..., *q*C*<sup>t</sup> n* is made up of the quantum trial individuals at the *t*th generation, where *n* is the size of the population.
