## 3 Results  
### Performance Analysis of Test Functions  
The Python code implements seven metaheuristic algorithms (QSA, ABC, GA, PSO, SA, CS and QPSO), with each algorithm being executed 10 times using random seeds to ensure independent runs. The final best fitness value from each run is recorded, and relevant statistical data (minimum, maximum, average, and standard deviation) are presented in Tables 1 and 2 for HyperSpehere and Levy functions, respectively. Convergence curves (average best fitness versus iteration) are also plotted in Figures 1 and 2 for HyperSpehere and Levy functions, respectively. In the context of the HyperSphere function, which is characterized as a unimodal and convex problem, on the 30-dimensional HyperSphere benchmark, the proposed **QSA** delivers an unequivocal performance lead. QSA's mean residual error is a mere **5.04**, whereas the closest challengers are more than twice as high--**GA** at 11.34, **PSO** at 11.35, and the quantum-inspired **QPSO** at 11.14--while classical meta-heuristics degrade sharply (**ABC** 60.15, **CS** 141.53, **SA** 192.35). Expressed in relative terms, QSA reduces the average error by approximately **55%** versus GA/PSO/QPSO, by **92%** versus ABC, and by more than **96%** versus CS and SA. Crucially, this dominance is consistent across the entire distribution of runs: QSA's _worst_ outcome (8.88) is virtually equal to GA's _best_ (7.14) and remains well below every competitor's mean. Moreover, QSA exhibits the tightest dispersion (\(\sigma=1.87\)), underscoring its stability. Combined with the highly significant ANOVA (\(F=948.23\), \(p<10^{-15}\)), these findings confirm that QSA not only attains the lowest error but also delivers the most dependable solutions on this challenging high-dimensional landscape. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Algorithm** & **Best** & **Worst** & **Average** & **Std** \\ \hline QSA & 2.8626 & 8.8837 & 5.0449 & 1.8697 \\ ABC & 44.1245 & 76.6285 & 60.1495 & 9.9259 \\ GA & 7.1431 & 17.4777 & 11.3425 & 2.8794 \\ PSO & 3.4040 & 33.9887 & 11.3507 & 9.5015 \\ SA & 143.3476 & 237.2534 & 192.3497 & 28.2362 \\ CS & 106.8212 & 159.7109 & 141.5257 & 15.8542 \\ QPSO & 6.1990 & 18.8431 & 11.1429 & 4.0122 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison of different optimization algorithms on the HyperSphere function (dim = 30, bounds = \([-5.12,5.12]\)). Figure 1: Convergence curves for the HyperSphere function using different optimization algorithms. The x-axis represents the iteration number, while the y-axis shows the best fitness value achieved at each step. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Algorithm** & **Best** & **Worst** & **Average** & **Std** \\ \hline QSA & 1.9176 & 3.9620 & 2.9017 & 0.7214 \\ ABC & 13.1845 & 35.5902 & 27.8299 & 5.9480 \\ GA & 1.8976 & 7.6294 & 4.8702 & 1.8090 \\ PSO & 4.1973 & 12.7546 & 7.2099 & 2.4210 \\ SA & 41.8375 & 61.5993 & 53.2947 & 6.2501 \\ CS & 23.0707 & 41.4601 & 34.7897 & 5.5807 \\ QPSO & 1.5833 & 5.4424 & 3.4059 & 1.0968 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance comparison of different optimization algorithms on the Lévy function (dim = 30, bounds = \([-5.12,5.12]\)). Figure 2: Convergence curves for the Lévy function using different optimization algorithms. The x-axis represents the iteration number, while the y-axis shows the best fitness value achieved at each step. The performance of QSA is further highlighted by the Levy function, which represents a more challenging multimodal landscape. On the 30-dimensional _Levy_ benchmark, the proposed **QSA** again proves to be the most reliable performer. Although **GA** attains a marginally smaller single best value (1.8976) than QSA's 1.9176, QSA dominates every aggregate metric that matters for practical optimization: its _mean_ error is only **2.90**, which is **40%** lower than GA (4.87), **60%** lower than PSO (7.21), and nearly an order of magnitude lower--about **90%**--than ABC (27.83), CS (34.79) and SA (53.29).