## 1 Introduction
The calculation results of most meta-heuristic algorithms are random due to the initial parameter settings, and there is a risk of falling into a local optimum when dealing with complex problems. On this basis, the combination of various meta-heuristic approaches is derived to solve multi-objective problems, such as butterfly algorithm-based clustering combined with ant colony optimization-based routing [22], improved duck and traveler optimization-based clustering combined with artificial gorilla troop optimization-based routing [23], hybrid gray wolf optimization, and the marine predator algorithm for clustering integrated with hybrid gray wolf optimization and the graph model for routing [24]. However, the complexity of the protocols is largely increased for execution. Therefore, hybrid meta-heuristic and traditional protocols are utilized to balance the overall performance of a network. For instance, the BAT algorithm for clustering combined with the fuzzy logic-based routing technique [14] adjusts the control parameters of fuzzy rules with the shuffled frog-leaping algorithm [17], and fuzzy logic model-based clustering was combined with particle swarm optimization-based routing [25]. In recent years, the universal applicability of quantum mechanics has been continuously proved, and the integration of quantum theory into intelligent optimization has emerged as an attractive research branch. Due to its ability to significantly enhance the solution speed compared to classical algorithms, scholars have incorporated concepts and mathematical mechanisms from quantum computing into existing optimization algorithms. For instance, they have integrated the concept of quantum computing into the traditional tunicate swarm algorithm to solve objective functions and obtained a clustering scheme that minimizes network energy consumption [26]. [PERSON] et al. combined quantum coding with a genetic algorithm and used qubits to represent candidate routes that explore and evaluate multiple routes simultaneously through quantum parallelism [27]. As a classical meta-heuristic algorithm, the simulated annealing (SA) algorithm is widely used in wireless sensor routing protocols. Despite being a random-step process, the al gorithm is prone to stay in a local optimum when the objective function has many high obstacles [28]. To overcome this problem, the quantum annealing (QA) algorithm was developed by replacing the temperature parameter of the SA algorithm with the intensity of the tunneling field. The quantum wave motion is generated through the mechanism of quantum tunneling, enabling the attainment of an optimal point across the potential barrier [29]. WSNs consist of a multitude of nodes and complex network topologies, and the network environment changes constantly over time. Consequently, it is imperative for algorithms to possess a certain level of adaptability. The quantum annealing algorithm demonstrates robust global search capability, effectively optimizing routing schemes. Its inherent parallelism enables efficient handling of large-scale problems and facilitates rapid convergence speed, thereby significantly enhancing routing efficiency and prolonging network lifespan. We utilized the accuracy of the fuzzy logic in optimization decisions to select CHs. Considering that Type-2 fuzzy rules require a large amount of computing time, we adopt the quantum annealing algorithm with a fast convergence rate in the routing stage. The proposed protocol in this paper aims to transmit data efficiently and effectively while also conserving energy and reducing computational time. In this paper, a hybrid of a fuzzy logic system and a quantum annealing algorithm-based clustering and routing protocol (FQA) is proposed to improve the stability of the network and extend the network lifetime.