## 1 Introduction
Their review listed Simulated Annealing (SA) as the third algorithm and fourth was The Particle Swarm Optimization (PSO) algorithm. The Artificial Bee Colony (ABC), Cuckoo Search (CS), and Quantum Particle Swarm Optimization (QPSO) algorithms were the other popular algorithms. As a result, we used SA, GA, PSO, ABC, CS, and QPSO for benchmarking and the TSP as our algorithm. We also evaluated the performance of these approaches on the continuous optimization problem using test functions. The SA algorithm was developed by [PERSON] et al. [24] as a general optimization technique, and it has since been applied to a range of combinatorial and continuous optimization problems. Inspired by metal-lurgical annealing, SA systematically lowers the \"temperature\" parameter to shift from exploratory random moves to opportunistic local improvements. Despite its simplicity, SA has shown remarkable performance on combinatorial problems such as circuit design, pattern recognition, and layout optimization [24; 25]. [PERSON] [26] developed the idea of GA and investigated how it might be used for trial allocation. According to [PERSON] [26], he finds inspiration in the fact that most creatures evolve through two fundamental processes. These are sexual reproduction and natural selection. While the latter allows genes to mix and recombine among offspring, the former selects which individuals in a population survive to reproduce. Genetic material is swapped when the sperm and egg unite, as chromosomes that match align and then partially cross over. A thorough study of genetic algorithms was carried out by [PERSON] et al. [27], who covered their historical evolution, present trends, and possible future developments. PSO algorithm was first proposed by [PERSON] and [PERSON] [28]. They took inspiration from social behavior of fish and birds. Each potential solution is represented as a \"particle\" in the search space, and this method optimizes solutions through group cooperation. Since the Kennedy and Eberhart till now, many variations of PSO have been introduced. A thorough review of PSO developments was given by [PERSON] et al. [29], who also discussed how well they work for various optimization issues. ABC is a swarm intelligence method inspired by the foraging behavior of honey bees [30; 31]. In ABC, candidate solutions are associated with food sources, and bees can be categorized into employed, onlooker, and scout types. Each plays a role in exploring and exploiting different regions of the search space. Bin packing, flow shop scheduling, and global numerical optimization are just a few of the continuous and discrete optimization issues in which ABC has proven to be successful [32]. The Discrete Cuckoo Search algorithm represents an enhanced adaptation of the original Cuckoo Search CS method, tailored for solving combinatorial optimization problems such as the Traveling Salesman Problem (TSP). By reconstructing the population structure and introducing a novel category of cuckoos, the authors demonstrated that DCS achieves superior performance on benchmark TSP instances from the TSPLIB, outperforming several conventional metaheuristic algorithms [33]. More recently, local minimization in large-scale discrete and continuous optimization problems has been seen from fresh angles thanks to quantum-inspired metaheuristics [34]. Approaches such as quantum-behaved PSO (QPSO) introduce wave-function based position sampling, which can stochastically \"jump\" through search spaces in a more diverse way than classical PSO [35]. Similarly, quantum annealing frameworks advocate the use of quantum tunneling to bypass high energy barriers more efficiently than purely thermal-based simulated annealing [36; 37]. Despite these advances, the majority of quantum-inspired algorithms are primarily targeted at continuous spaces or specialized discrete mappings, and they often lack robust multi-agent collision or repulsion strategies to maintain the diversity of the population in combinatorial landscapes [38].