## 2 Materials and Methods
### Definition of Quantum Layers  
The definition of quantum layers in the model is carried out to perform quantum operations and integrate them with classical learning structures. Quantum layers are defined through functions such as H_layer, RY_layer, and entangling_layer and are used in creating PQCs. These layers calculate data through quantum circuits, integrate quantum information processing features into the model, and offer a perspective distinct from classical learning methods. The model operates by creating superposition, performing parametric transformations, and conducting entanglement processes, all of which collectively expand the model's learning capacity. Figure 3 visually represents these processes, explaining each stage of the quantum circuit step by step. Figure 3: Illustrates the stages of superposition, entanglement, and measurement in a quantum circuit. The numbers 0, 1, 2, and 3 represent the qubits in the circuit. The “H” denotes the Hadamard Gate, “RY” represents the Rotation-Y Gate, and the circuit also includes CNOT Gates and measurement symbols. The Hadamard gate (H gate) plays a fundamental role in quantum computation by placing qubits into a superposition state, where they simultaneously exist in both 10 and 11) states. This allows quantum systems to process multiple possibilities in parallel, vastly increasing computational efficiency ([PERSON] & [PERSON], 2010). The RY gate, on the other hand, rotates qubits around the \(Y\)-axis of the Bloch sphere based on specific angles, enabling the representation of more complex quantum states and enhancing the model's ability to analyze diverse data features. Entanglement, achieved through CNOT gates, establishes a quantum correlation between qubits, ensuring that the state of one qubit is dependent on the state of another. This interconnectedness forms the basis for processing complex relationships in quantum systems. Together, these concepts--superposition, rotation, and entanglement--constitute the foundational principles of quantum computation and enable the creation of highly efficient learning models [23]. The computation process begins by applying Hadamard transformations to all quantum bits. This operation places the qubits simultaneously in both 10) and 11) states, creating superposition. Mathematically, this is expressed as follows:  
\[ \begin{array}{l}\mathrm{H}\,|\,0\rangle=1/\surd{2}\,(10)+11),\\ \mathrm{H}\,|\,1\rangle=1/\surd{2}\,(10)-11)).\end{array}  \tag{1}\]  
For example, Hadamard gates are applied sequentially to Qubit 0, Qubit 1, Qubit 2, and Qubit 3. Superposition enables parallel computation in quantum information processing, allowing the model to learn more complex data relationships. These transformations are performed on each qubit using the qml.Hadamard(wires = idx) command. This stage of superposition expands the model's initial knowledge base, providing the fundamental structure needed for subsequent operations [24]. Following the Hadamard transformation, RY transformations are applied using angles derived from the data. Mathematically, the RY gate is represented as follows:  
\[\mathrm{RY}(\theta)\,|\,\psi\rangle=\cos(\theta/2)\,|\,\psi\rangle+\sin(\theta /2)\,|\,\psi\rangle. \tag{2}\]  
This operation rotates the state of each qubit around specified angles, enabling data analysis in various directions. For instance, RY gates are applied to Qubit 0, Qubit 1, Qubit 2, and Qubit 3 sequentially, and these transformations are performed using the qml.RY function. The angles for RY transformations are determined based on data from q_input_features. These angles are optimized during the learning process to adapt to the data, contributing to the model's ability to evaluate data features from different perspectives. These parametric transformations enhance the model's flexibility, enabling a deeper learning process [25]. After RY transformations, entanglement operations are performed. At this stage, dependencies between qubits are established using CNOT gates. Mathematically, the CNOT gate operates as follows:  
\[