## 4 The Proposed Method  
Unlike the traditional QGA [11], our proposed FbQGA for Feature Selection consists of three main phases, as depicted in Figure 6. The steps outlined with red contours represent the new enhancements. 1. **Initialization:** This step remains the same as in the classic QGA, where a population of Q-bit individuals is randomly generated and then converted into binary individuals. We use an SVM classifier to compute the fitness values. 2. **Adaptive Rotation:** Typically, a quantum rotation operation involves two key components: the rotation direction and the rotation angle. The details of each component are provided below in this paper. * **Rotation direction:** A quantum individual undergoes rotation only when the following conditions are met: the quantum individual is inferior to the global best solution \(b^{t-1}\), and the corresponding binary digit of the individual differs from \(b_{i}^{t-1}\). This mechanism is illustrated in Figure 7. Here, \(f\) represents the fitness function, \(x_{ij}\) is the binary value of the \(j^{th}\) feature of the \(i^{th}\) individual, and \(b_{j}\) refers to the binary value of the global best solution. * **Rotation angle:** we propose a novel adaptive rotation angle that calculates the magnitude of the rotation angle for each Q-bit of each quantum individual in the Equation (8) \[\Delta\theta_{ij}^{t}=\theta_{min}+\frac{log(1+\frac{tf_{j}^{t-1}}{n+\#})}{ \frac{1}{avg.fitness^{t-1}}}\] (8) Here, \(\Delta\theta_{ij}^{t}\) represents the rotation angle of the \(j^{th}\) Q-bit in the \(i^{th}\) Q-bit individual at generation \(t^{th}\). \(\theta_{min}\) is the minimum rotation angle, set to \(0.0025\pi\). The population size and the total number of features are denoted by \(n\) and \(m\), respectively. The feature frequency vector, \(ff^{t-1}\), is shown in Figure 8 and is calculated as the sum of all binary individuals. Specifically, \(ff_{j}^{t-1}\) represents the frequency of the \(j^{th}\) feature appearing in the binary population at the previous generation \((t-1)^{th}\). The term \(avg.fitness^{t-1}\) refers to the average fitness of the population at the \((t-1)^{th}\) generation. The core concept is that different features have different levels of importance. To evaluate their contributions, we measure the frequency of their occurrence in the population and examine how this frequency influences the population's  
Figure 5: Polar plot of the rotation gate for Q-bit individuals. fitness. To avoid bias due to variations in population size (\(n\)) and the total number of features (\(m\)), we normalize the results using these factors. As the population's fitness improves or worsens, the rotation angle adjusts accordingly, increasing or decreasing in response. Similarly, when the frequency of a feature changes, the rotation angle for that feature adjusts proportionally to the change in fitness. Figure 6: The architecture of Adaptive Feature-based Quantum Genetic Algorithm (FbQGA) Feature Selection (_The steps in red-contour are the new enhancements_). * **Adaptive Mutation:** We employ the adaptive mutation rate based on the population diversity [20]. The population diversity is measured through the standard deviation of the population's fitness values, as described in Equation (9). \[st{d^{t}}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(fit(x_{i}^{t})-fit(\overline{x^{t}}))^{ 2}}\] (9) Here, \(fit(\overline{x^{t}})\) represents the average fitness of the \(t^{th}\) generation, and \(fit(x_{i}^{t})\) is the fitness value of individual \(x_{i}\) in the \(t^{th}\) generation. As the standard deviation decreases, the mutation rate is increased to promote greater population diversity. The adaptive mutation rate is given by Equation (10). \[P_{m}^{t}=P_{f}*(1+\frac{fit_{max}^{t}-st{d^{t}}}{fit_{max}^{t}+st{d^{t}}})\] (10) In this equation, \(P_{f}\) is a fixed mutation rate, and \(fit_{max}^{t}\) is the highest fitness in the \(t^{th}\) generation. The adaptive mutation rate \(P_{m}\) is influenced by both the fixed mutation rate and the population's diversity.