## 3 Results
### Performance Analysis of The Traveling Salesman Problem  
Table 3 presents a concise overview of the experimental findings, wherein QSA is juxtaposed with four alternative metaheuristics (namely ABC, GA, PSO, SA, CS, and QPSO) in the context of a 200-city TSP. Each algorithm was permitted to execute until either convergence was attained or a predefined iteration limit was reached. The optimal route distance and its corresponding wall-clock time are documented.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Algorithm** & **Best Distance** & **Elapsed Time (s)** \\ \hline QSA & 6609.61 & 0.84 s \\ ABC & 8395.93 & 0.36 s \\ GA & 9247.81 & 0.61 s \\ PSO & 9690.38 & 6.68 s \\ SA & 9973.85 & 0.20 s \\ QPSO & 8147.23 & 0.26 s \\ CS & 9626.59 & 5.71 s \\ \hline \hline \end{tabular}
\end{table}
Table 3: Performance comparison of different optimization algorithms on the 200-city TSP.  
As shown in Table 3, **QSA** delivered the most compact tour while maintaining competitive execution time. QSA attained a best-of-run distance of **6609.61**, outperforming every comparator by a wide margin: its tour is **21.3% shorter than ABC** (8395.93), **28.5% shorter than GA** (9247.81), **31.8% shorter than PSO** (9690.38), **33.7% shorter than SA** (9973.85), **18.9% shorter than the quantum-inspired QPSO** (8147.23), and **31.3% shorter than CS** (9626.59). From a computational effort perspective, QSA required only **0.84 s** to reach its best solution, placing it firmly in the sub-second regime shared by most heuristic baselines and rendering it nearly an order of magnitude faster than PSO (6.68 s). Although ABC, SA, and QPSO terminated slightly sooner (0.36 s, 0.20 s, and 0.26 s, respectively), the dramatic reduction in tour length achieved by QSA more than compensates for this modest time overhead. In practical terms, QSA offers _the best distance-time trade-off:_ it is the only solver that combines sub-second convergence with a tour length at least one-fifth shorter than any competitor. These findings reinforce our continuous-domain results and underscore QSA's versatility as a high-quality, time-efficient meta-heuristic for large-scale discrete optimisation.  
Figure 3 shows convergence curves for different optimization algorithms on the TSP function that represents the iteration number, while the best fitness value achieved at each step.  
Figure 4 provides a visual representation of the \"best route\" solution obtained by each algorithm (QSA, ABC, GA, PSO, SA, QPSO, and CS) for the 200-city TSP instance. While initially, each of the resulting tours appears visually dense with numerous intersecting edges, two key observations can be made. Upon comparison of the plots, it is evident that subtle variations in edge lengths and angles collectively exert a significant influence on the total route distance. Modifications in the sequence of visited cities can result in a reduction in excessively elongated segments that extend across the entirety of the 100x100 area. Notably, QSA's solution, despite its intricate nature, attains the lowest overall distance through the minimization of extended \"detours\" and the balancing of local city clusters. Moreover, visually, all tours appear to be equally \"tangled\" due to the necessity of connecting 200 points without overlaps in visitation. However, an algorithm that achieves a shorter total distance is likely to have fewer extremely long edges that span diagonally across the region. For instance, a concentration of smaller segments or more orderly sub-loops suggests the algorithm's capacity to exploit local proximity effectively.  
Figure 3: Convergence curves for different optimization algorithms on the TSP function. The x-axis represents the iteration number, while the y-axis shows the best fitness value achieved at each step.