# Quantum Selection for Genetic Algorithms Applied to Electromagnetic Design Problems
## 7 Results and Discussion
### Final Results Statistics  
The statistics of the final results for all the tests are presented as box plots in Figure 7. A box plot is a simple plot that allows the visualization of the results of an experiment in a concise way. Its four main parts have the following interpretation in the context of an evolutionary optimization algorithm:  
* Median: It is the red line inside the box; it is similar to the mean, which has already been interpreted in the previous subsections. Its 95 % confidence interval is given by the black lines above and below it.
* Interquartile range (IQR): It is a box that represents the distance between the first and third quartiles of the data. Its interpretation is linked to the repeatability of the results. A smaller IQR means that the algorithm is less inclined to have an unexpected  
Figure 6: Convergence speed of the mean fitness value for all 200 tests performed per algorithm, using both quantum circuit measurement techniques. The benchmarks include (**a**) Manhattan norm, (**b**) Rastrigin function, (**c**) Griewank function, (**d**) Microstrip filter, and (**e**) electromagnetic absorber. Displayed curves include both the instantaneous square root of the rate of fitness change and a 10-sample moving average. Colored backgrounds correspond to convergence phases: EGs (early generations), MGs (middle generations), and LGs (late generations) for mathematical functions, and IGs (initial generations) and FGs (final generations) for design problems. Higher values indicate faster convergence.  
behavior. On the other hand, a large IQR makes the algorithm less reliable on a single run.
* Whiskers: They contain all the data 1.5 IQR in distance from the box. In this case, their interpretation is the same as the IQR.
* Outliers: They are all the data not contained in the whiskers; in this work, all outliers are preserved. Depending on their position, they have the following interpretation:
* High-fitness outlier: If there are many, they suggest that the algorithm has a tendency to have low-performing executions, usually due to stagnation.
* Low-fitness outlier: If there are many, they suggest that the algorithm has a tendency to have higher-than-average performing executions.  
In every case, the performance of both versions of GAQS outperforms GA in both median and IQR, making them a more reliable option. Furthermore, the presence of high-fitness outliers is almost exclusive of GA, except for the test done with the Griewank function in GAQS. Moreover, the presence of low-fitness outliers is limited to GAQS SBX, showing that the strong exploration of SBX may result in flukes. However, these are not representative of the real performance of the algorithm.  
The best-performing algorithm is problem-dependent. Nonetheless, for MTFs, GAQS is more likely to be the best, while for the considered EMDP, GAQS SBX is more likely to be  
Figure 7: Box plots of the final fitness values obtained across 200 runs for each algorithm and quantum measurement method are shown for (**a**) Manhattan norm, (**b**) Rastrigin function, (**c**) Griewank function, (**d**) Microstrip filter, and (**e**) electromagnetic absorber. Lower medians, smaller interquartile ranges (IQRs), and fewer high-fitness outliers indicate more consistent and effective optimization.  
the best. In any case, the algorithm with the best performance in terms of median is usually the best algorithm in IQR too. Furthermore, the IQR is independent of the measuring methodology used, while the median is affected by it.