## 1 Introduction
In this paper, we propose a modified CH selection method by assigning an individual weight to each sensor node based on its node degree, the average distance between the CH and its member nodes, as well as this, we assign residual energy as a function of transmission power. As mentioned, our work focuses on a homogeneous, randomly connected network, and our goal is energy-efficient CH selection by controlling the transmission power. Two nodes can communicate directly if their transmission ranges overlap, which signifies the connectivity between them. For multi-hop communications, connec  
Figure 1: Graphical abstract (3D) of clustering architecture in WSN. tivity is a fundamental property, but because we are concerned with finding the optimum transmission range, we must consider a fully connected network with a minimum node degree (i.e., 1-connectivity). A network with k-connectivity (k \(\geq\) 2) has much better fault tolerance than a network does with only 1-connectivity, but higher connectivity requires more power consumption [22]. Therefore, a fully connected network with a minimum node degree (1-connectivity) is believed, in this paper, to lessen the energy consumption. During intra-cluster communications, each node adjusts its transmission power based on the cluster range or radius, instead of transmitting at maximum power. This ensures that the nodes are restrained from unwanted power usage, which reduces the amount of battery drainage. However, for fast data transmission from the CH to the BS, the maximum amount of transmission power is needed, which resembles single-hop communications. In a communication model, there exist several problems which can be classified into their hardness on different categories. These problems can be categorized into four complexity classes as: P (easy), NP (medium), NP-complete (hard), and NP-hard (hardest). The main concept of the complexity is how efficiently an algorithm can solve a problem [23]. Hierarchical clustering algorithms are based on classical algorithms in which several clusters are formed [24; 25]. If the number of clusters and the total number of elements are represented by k and n, respectively, then the time complexity of a hierarchical algorithm is O(kn\({}^{2}\)). This time complexity represents time complexity that is similar to that of NP-hard (non-deterministic polynomial) problems [26]. Hence, CH selection is generally recognized as an NP-hard optimization problem [27; 28], and its time complexity is equivalent to O(kn\({}^{2}\)) if the clustering problem is solved by a classical approach. As a result, the CH selection procedure is also time consuming. Therefore, the main challenge is to overcome time complexity, which incurs huge computational and data processing times. In order to improve on the time complexity problem, quantum algorithms can play significant roles. Recently, CH selection was proposed via the Quantum Approximate Optimization Algorithm (QAOA) [29]. The QAOA structure is classified into two parts: parameterization and classical optimization. A parameterized quantum circuit consists of building a Hamiltonian problem where the proper optimization of the parameters is essential [30]. In a parameterized quantum circuit, the components of the circuit are demonstrated by \(\stackrel{{\rightarrow}}{{\delta}}\) and the output state \(|\Phi(\stackrel{{\rightarrow}}{{\delta}})\rangle\). The Quantum Max Cut problem solved by the QAOA needs to optimize the circuit parameters efficiently [31]. The main problem of the QAOA algorithm is that it considers each qubit (qi) as an individual node, which unintentionally limits it from forming clusters in the whole network. For instance, IBM's 127-qubit Eagle is the biggest quantum computer, yet it has only 127 qubits. Consequently, the highest number of network nodes that can be used is only 127. For a larger network (>127), the above-mentioned algorithm is impractical. To overcome this issue, we present a new CH selection technique based on the Quantum Search Algorithm (QSA) using weighted targets [32; 33].