## 4 Discussion  
QSA is introduced as a new algorithm to the literature by combining collision-based diversity preservation, quantum-inspired tunneling, overlap-based partial solution sharing, and local optimization to address a wide range of continuous and discrete optimization problems. QSA is evaluated on multiple test functions, such as Levy and HyperSphere in the continuous domain, and on a large-scale TSP in the discrete domain. As indicated in Table 4, [PERSON] and [PERSON] [23] conducted a detailed review of metaheuristic approaches for TSP-based scheduling optimization challenges. We compared ABC, GA, PSO, SA, CS, and QPSO with our novel algorithm QSA. So, we used these metahuristics to compare our method. \begin{table}
\begin{tabular}{l c} \hline \hline
**Algorithm** & **Total Publications** \\ \hline Genetic Algorithm & 2257 \\ Ant Colony Optimization & 1338 \\ Simulated Annealing & 775 \\ Particle Swarm Optimization & 468 \\ Tabu Search & 330 \\ Memetic Algorithm & 150 \\ Variable Neighborhood Search & 143 \\ Artificial Bee Colony & 98 \\ Differential Evolution & 96 \\ Firefly Algorithm & 59 \\ Genetic Programming & 53 \\ Clonal Selection Algorithm & 36 \\ Gene Expression Programming & 33 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Impact on the TSP (Total Publications) [23]. Figure 4: Visualization of the TSP routes obtained by different optimization algorithms. Each red dot represents a city, and blue lines indicate the travel paths connecting them. The seven subfigures correspond to different algorithmic approaches applied to the 200-city TSP instance. We compared the most widely used baseline optimization algorithms in the literature, ABC, GA, PSO, SA, CS, and QPSO, with our new algorithm. QSA in our testing was shown to consistently obtain low function values (around the global minimum) in fewer iterations. In experiments, QSA reached the best value with an average fitness value of 5.0449 on the HyperSphere benchmark, showing robust exploitation with enough exploration. Moreover, QSA's quantum tunnelling enabled it to avoid local troughs more efficiently than the other algorithms which are swarm or evolutionary algorithms on the Levy benchmark. Levy is a deceptively multimodal terrain. QSA outperformed the others with an average fitness value of 2.9017 on the Levy benchmark. In QSA, the incorporation of solutions within a continuous auxiliary space, in conjunction with the implementation of repulsive forces, was demonstrated to effectively mitigate the occurrence of premature clustering, a prevalent limitation of other algorithms such as GA and PSO. The collision-based dynamics of the algorithm were shown to impede the convergence of the search process on local minima, thereby fostering the exploration of more extensive search spaces and enhancing the efficacy of the global search process. Moreover, early iterations leveraged higher thermal energy to sample the space widely, while decaying \(\sigma\) values gradually reinforced tunneling around high-value basins. These elements worked together to offer QSA consistently low function values and rapid convergence. The TSP performance showed that QSA required a considerable amount of wall-clock time, but it produced the shortest route distance with a value of 6567.58 in our experiment. The mapping of permutations into a continuous 2D space allowed the collision and tunneling stages to function consistently. Following position updates, solutions were remapped to permutations via nearest-route or partial reinsertions, thereby allowing QSA to combine \"snowflake\" diversity and quantum leaps effectively in a purely combinatorial context. The superposition stage of QSA inserts quality sub-segments from the best solution into others, thereby accelerating the spread of partial \"good routes\". The subsequent 2-Opt-based local search further pruned unnecessary detours. It is evident that, collectively, these operators enhanced the algorithm's exploitation capacity, whilst collisions ensured that variety was maintained in possible tours.