## 2 Related Work  
Among the clustering algorithms, the energy-efficient clustering scheme (EECS) [20] was proposed for effective topology control in wireless sensor networks. Under this protocol, the nodes select a CH at the minimum distance for cluster formation. The CH selection procedure is based on the probability of a certain threshold value between 0 and 1, which is a random process that is similar to the that of the LEACH protocol [35]. A candidate node that is going to become a CH needs to ensure that its residual energy is within radio range \(R_{compete}\). This radio range is inversely proportional to the square root of \(k_{opt}\), which is the optimal range of the CH. Under EECS, the optimal range selection equation is taken from the LEACH protocol, and there is no relation to transmission power for \(R_{compete}\). Therefore, the control of optimum transmission power in the whole network is somehow overlooked. To resolve this problem, the selection of an optimal number of cluster heads is proposed in this work, while managing optimum transmission power. On the other hand, in the EECS cluster formation phase, a complex cost function for distance prioritizes only the cluster-head-to-BS distance, which eventually increases the overhead and time complexity in the whole network. The total control of the given overhead complexity is O(N), and to overcome this, we propose a quantum algorithm for CH selection where the overhead complexity is reduced to O (\(\sqrt{N}\)). Power-Efficient Gathering in Sensor Information Systems (PEGASIS) [36] is an improvement over the LEACH protocol. The fundamental idea behind PEGASIS is for each node to communicate with and send data to its immediate neighbors, while they take turns acting as the leader, transmitting to the BS. The nodes arrange themselves into a chain by using a greedy algorithm. The data transmission is direct between the leader and the BS within a fixed transmission range. However, an adjustment of the transmission power level in the whole network is not discussed. Energy-efficient unequal clustering (EEUC) [37] was proposed for multi-hop routing. This clustering technique is based on competition range \(R_{comp}\) as a function of the distance to the BS. As the node moves farther from the base station, its competition range becomes narrower. Consequently, reduced cluster sizes are anticipated for the clusters that are closer to the base station. Although this protocol is efficient in intra-cluster communication, inter-cluster communication increases the energy overhead complexity O(N) due to multi-hop routing. Energy-Efficient Cluster Formation (EECF) [38] is known for being a distributed clustering protocol that considers the node degree and residual energy for cluster head selection. There is no mention of re-adjusting the transmission power level during each round, but a fixed transmission range for at least one cluster head cannot guarantee efficient communications. Like EECS, EECF has a worst-case algorithmic complexity of O(N) at each node. The Hybrid Energy-Efficient Distributed (HEED) [8] protocol considers CH selection based on the ratio of the estimated current residual energy in a sensor node and its maximum energy. Transmission power level control can be optimized by setting one specific cluster power level for intra- and inter-cluster communications. The connectivity requirement setting cluster range \(\text{R}_{\text{t}}\sim\sqrt{\frac{\log n}{n}}\) (where \(n\) is the total number of nodes) is for a unit square region in intra-cluster communications. However, the probability of connectivity in the whole network does not consider any specific limit on the transmission range, and no detailed explanation is given. Node density (\(\rho\)) and node degree (d) play essential roles in establishing connectivity in the network, but under the HEED protocol, it is somewhat questionable to not consider those parameters.