## 2 Related Works
Subsequently, the IMFO (improved moth flame optimization) algorithm was adopted to plan the shortest transmission path from CHs to the BS, including factors such as residual energy of CHs and distance among CH, relay CH, and BS. Compared with some existing protocols, OAFS-IMFO achieves significantly higher packet throughput while consuming more energy. The proposed hybrid differential evolution and simulated annealing (DESA) algorithm in [33] aims at maximizing the network lifetime of the WSN by combining the simulated annealing (SA) approach for global optimal solution with the DE approach for local search. The DESA algorithm takes only two factors into account, energy and distance between nodes, in the process of selecting cluster heads, which is not comprehensive enough. As a classic meta-heuristic algorithm, the SA algorithm has been widely applied in WSNs. However, it still has several limitations, such as easily falling into local optima. With the rapid development of quantum technology, scholars have integrated the concept and mathematical mechanism of quantum computing into existing optimization algorithms to improve their ability to search the solution space. Among these approaches, the quantum annealing algorithm swiftly emerged as a classical discrete quantum technology in quantum optimization algorithms. Compared to the simulated annealing algorithm, it can solve the NP-hard problem in WSNs more effectively. In [34], the clustering objective function was reformulated as a quadratic unconstrained binary optimization (QUBO) problem, and the paper proposed two encoding methods. One is the one-hot encoding, which associates each data point with a k-bit string, but this method fails to utilize quantum bits effectively and has inherent limitations; the other is binary encoding, which represents each data point's classification by a binary string and is executed hierarchically. The proposed method needs to be implemented with the help of commercially available quantum annealing hardware and a purely classical solver, \"qbsolv\" released by D-Wave. In [29], the mobile sink node collects data from all nodes within the communication range along the planned path and returns to the initial location after completing one round. The problem of finding the shortest path for a mobile sink node is essentially a traveling salesman problem (TSP). [PERSON] et al. employed quantum encoding techniques to represent the mobile path and utilized the Hamiltonian of the QA algorithm to guide the operation of a quantum revolving gate which effectively addresses the energy hole problem. With the excellent performance of fuzzy theory in optimizing decisions and reducing energy consumption in WSNs, fuzzy logic is widely used in selecting CHs and clustering. Utilizing the fuzzy logic control system enables real-time adjustments based on dynamic environmental conditions, thereby enhancing the fault tolerance and stability of the system. Combining fuzzy logic and meta-heuristic algorithms can fully leverage their respective strengths and enhance capabilities in addressing intricate optimization problems. The generation of fuzzy rules in [13] involves considering energy and utility factors, and the fuzzy inference system is employed to identify relay nodes. The minimum length route is determined based on coverage distance, energy, utility factor, and bandwidth availability. [PERSON] et al. [14] proposed load-balanced fuzzy C-means (FCM) algorithm-based clustering integrated with BAT approach-based routing (FC-RBAT). They used FCM to form clusters and find the centroid of the clusters. Then, they selected proper CHs among the sensor nodes from each cluster, which constructed the objective function based on residual energy, intra-cluster distance, distance to cluster centroid, and distance to the BS. The routing process is conducted using a BAT-inspired approach that considers the lifetime of CHs, the number of hops, and total transmission distance as essential parameters.