# Quantum Selection for Genetic Algorithms Applied to Electromagnetic Design Problems
## 7 Results and Discussion
### Fitness's Mean Value Evolution  
The mean value evolution per algorithm is presented in Figure 5 for all the considered tests. In general, for the MTF (Figure 5a-c), three distinct regions can be defined using the differences between the fitness values of the algorithms. The first region, called early generations (EGs), goes from generation 1 until GAQS surpasses GA. The second region, middle generations (MGs), goes from GAQS surpassing GA to GAQS equaling GAQS SBX. The last region, called late generations (LGs), goes from GAQS equaling GAQS SBX onward. In simple equations,  
* Early generations (EGs): \(\mathrm{fitness_{GAQS\;SBX}}\leq\mathrm{fitness_{GA}}\leq\mathrm{fitness_{GAQS}}\);
* Middle generations (MGs): \(\mathrm{fitness_{GAQS\;SBX}}\leq\mathrm{fitness_{GAQS}}\leq\mathrm{fitness_{GA}}\);
* Late generations (LGs): \(\mathrm{fitness_{GAQS}}\leq\mathrm{fitness_{GAQS\;SBX}}\leq\mathrm{fitness_{GA}}\). In EG, the mechanisms introduced in GAQS for increasing genetic diversity slow down the initial evolution of the algorithm compared to GA. However, the strong exploratory characteristics of SBX overcome the initial slowness of QSO, giving GAQS SBX the best overall performance. In MG, the higher genetic diversity conferred by QCO in GAQS eventually overcomes GA. However, due to the convex nature of the Manhattan norm and the central minimum in all considered non-convex MTFs, heuristic crossover starts to perform better than SBX until GAQS surpasses GAQS SBX. In LG, the advantage of the heuristic crossover becomes clear as SBX causes a decrease in the convergence speed. Moreover, the measuring method's impact is problem-dependent. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material Name** & **Relative Electric Permittivity (\(\epsilon_{r}\))** & **Loss Tangent** \\ \hline \(M_{1}\) & \(2-j0.1\) & \(0.050\) \\ \(M_{2}\) & \(3-j0.3\) & \(0.100\) \\ \(M_{3}\) & \(7-j0.4\) & \(0.057\) \\ \(M_{4}\) & \(10-j0.6\) & \(0.060\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dielectric characteristics of the materials used for the multilayer electromagnetic absorber. Overall, quantum computers are slower than Aer simulators, and noisy measurements worsen GAQS SBX performance, especially in EG. The timing of EG, MG, and LG depends entirely on the MTF. For the EMDP, only two regions can be observed: initial generations (IGs) and final generations (FGs). They are equivalent to EG and MG and they are defined as  
* Initial generations (IGs): fitness\({}_{\text{GAQS SBX}}\leq\) fitness\({}_{\text{GAQS}}\leq\) fitness\({}_{\text{GAQS}}\);
* Final generations (FGs): fitness\({}_{\text{GAQS SBX}}\leq\) fitness\({}_{\text{GAQS}}\leq\) fitness\({}_{\text{GA}}\). Due to the low number of generations, LG is not observed. It might appear in future generations near the minimum, where performance converges. Thus, GAQS may eventually outperform GAQS SBX. For the microstip filter in Figure 5d, the IG shows fast convergence for GAQS SBX due to its strong exploratory capacity, GA stagnates after generation 25, and in FG, GAQS SBX shows rapid improvement followed by GAQS; after generation 100, GAQS SBX slows down. Figure 5: Evolution of the mean fitness value across 200 tests for each algorithm, using both quantum circuit measurement techniques. The following benchmark problems are presented: **(a)** Manhattan norm, **(b)** Rastrigin function, **(c)** Griewank function, **(d)** Microstrip filter, and **(e)** electromagnetic absorber. The shaded backgrounds denote convergence phases defined as EGs (early generations), MGs (middle generations), and LGs (late generations) for mathematical functions, and IGs (initial generations) and FGs (final generations) for electromagnetic design problems. Lower fitness values are preferable. All individual optimization runs and the confidence interval of the means are available in Appendix A. On the other hand, GA tends to stagnate after generation 25 (see Appendix A), negatively affecting its performance.