## 3 Materials and Method
### Quantum Genetic Algorithms  
Quantum Genetic Algorithm (QGA) for feature selection are enhanced versions of traditional genetic algorithms that incorporate principles from quantum computing, though they do not require actual quantum hardware. In quantum computing, information is represented using quantum bits, or qubits. Unlike classical bits, which exist strictly as either '0' or '1', qubits can exist in a superposition of both states simultaneously. The mathematical representation of a qubit is given by Equation (1). \[|\Psi\rangle=\alpha|0\rangle+\beta|1\rangle \tag{1}\]  
In this representation, \(\alpha\) and \(\beta\) are complex numbers that denote the probability amplitudes of their respective states. The probability of the qubit being in the \"0\" state is given by the squared magnitude of \(\alpha\), represented as \(|\alpha|^{2}\), while \(|\beta|^{2}\) represents the probability of the qubit being in the \"1\" state. To maintain normalization, Equation (2) ensures that the total probability across all possible states sums to one. \[|\alpha^{2}|+|\beta^{2}|=1 \tag{2}\]  
In the context of feature selection using QGA, \(|\alpha^{2}|\) and \(|\beta^{2}|\) correspond to the probabilities of a feature being unselected (state \"0\") and selected (state \"1\"), respectively. As these probabilities approach either 1 or 0, the Q-bit individual becomes more deterministic, ultimately converging to a single state and gradually reducing diversity. This characteristic enables the Q-bit representation to balance both exploration and exploitation simultaneously [11]. A Q-bit individual, structured as a sequence of Q-bits, is mathematically defined in Equation (3). \[[\frac{\alpha_{1}}{\beta_{1}}|\frac{\alpha_{2}}{\beta_{2}}|\cdots|\frac{ \alpha_{m}}{\beta_{m}}] \tag{3}\]  
where \(|\alpha_{i}^{2}|+|\beta_{i}^{2}|=1\); \(i=1,2,\ldots,m\). Figure 4: Genetic Algorithms mutation process. A quantum genetic algorithm for feature selection encompasses the below essential steps. * **Initialization:** A quantum chromosome population is randomly generated. \(Q(t)=\{q_{1}^{t},q_{2}^{t},\ldots,q_{n}^{t}\}\) at generation \(t\), where \(n\) is the size of population, and \(q_{i}^{t}\) is a Q-bit individual defined as Equation (4). \[q_{i}^{t}=[\frac{\alpha_{i1}^{t}}{\beta_{i1}^{t}}\frac{\alpha_{i2}^{t}}{\beta_ {i2}^{t}}|\cdots|\frac{\alpha_{im}^{t}}{\beta_{im}^{t}}]\] (4) The number of features is represented by \(m\). After that, in the act of observation, \(n\) Q-bit individuals are converted into (or collapsed) \(n\) binary individuals (\(X(t)=\{x_{1}^{t},x_{2}^{t},\ldots,x_{n}^{t}\}\)). One binary solution; \(x_{i}^{t},i=1,2,\ldots,n\); is a binary string of the length \(m\), and is formed by selecting each bit using the probability of q-bit following the rules below Equation (5). \[if|\alpha_{i,j}^{2}|<rand[0,1];then\ x_{i,j}=0\ else\ x_{i,j}=1\] (5) where \(i\) represents \(i^{th}\) individual, \(j\) is the \(j^{th}\) feature, and \(rand[0,1]\) generates a random number between 0 and 1. The fitness function, also known as the evaluation function, is designed to assess the quality of each quantum feature subset. In feature selection problems, whether a lower or higher fitness value indicates a more optimal solution depends on the objective function. If the objective function is a minimization problem (e.g., minimizing error or redundancy as in Equation (6)), a lower fitness value is preferable. Since fitness functions are crucial for evaluating candidate solutions during optimization, various methods have been proposed in the literature. These include evaluation metrics such as accuracy, F1 score, precision, and recall, as well as information-theoretic measures, error or loss functions, and linear combinations of subset cardinality with evaluation or loss metrics. An example of a fitness function is given in Equation (6). \[fitness=(\lambda*err)+(1-\lambda)*(\frac{f_{s}}{f_{T}})\] (6) Here, \(\lambda\) acts as a weighting factor for \(err\) relative to the size of the feature subset.