# A Reconfigurable Framework for Hybrid Quantum-Classical Computing
## 3 Proposed Framework
### Quantum Machine Learning Task
#### 3.1.2 Quantum Circuit Generation  
After the data pre-processing stage, the prepared data are to be used in the quantum circuit generation stage. In this stage, a trainable, parameterized quantum circuit representing the machine learning model will be constructed using the prepared data as the circuit's parameters. This quantum circuit is the quantum equivalent of a classical neural network, and the circuit parameters are equivalent to the network's weights. The circuit architecture is the conventional RealAmplitudes circuit architecture from [PERSON]'s circuit  
Figure 2: QML workflow for a quantum machine learning task.  
library [31], which is tailored for quantum machine learning tasks using real-valued parameters. The circuit structure incorporates an \(n\)-qubit system with \(L=5\) variational layers. This can be represented by Equation (1), where \(|\psi\rangle\) denotes the final quantum state, \(U_{i}(\theta_{i})\) represents the \(i\)-th variational layer with trainable parameters \(\theta_{i}\) (for \(i\) from \(1\) to \(L\)), \(U_{enc}(x)\) is the feature encoding layer for input data \(x\), and \(|0\rangle^{\otimes n}\) is the initial state of the \(n\) qubits. The circuit begins with the initial state \(|0\rangle^{\otimes n}\), representing an \(n\)-qubit quantum register initialized to the zero state. The operator \(U_{enc}(x)\) is the data encoding unitary, which maps the classical input \(x\) into a quantum state. Following this, a sequence of \(L\) parameterized unitary operations \(\{U_{1}(\theta_{1}),U_{2}(\theta_{2}),\ldots,U_{L}(\theta_{L})\}\) is applied, where each \(U_{\ell}(\theta_{\ell})\) represents a trainable unitary gate or block with associated parameters \(\theta_{\ell}\). These operations collectively form the variational layers of the quantum model. The composition of these gates yields the final state \(|\psi\rangle\), which is typically measured to extract information relevant to the task at hand.  
\[|\psi\rangle=U_{L}(\theta_{L})\cdot...\cdot U_{2}(\theta_{2})\cdot U_{1}( \theta_{1})\cdot U_{enc}(x)|0\rangle^{\otimes n} \tag{1}\]  
The circuit architecture consists of the following:  
* Feature mapping layer:
* Initializes qubits with Hadamard (H) gates for superposition.
* Applies RY rotations for data encoding.
* Parameters encode input features into quantum states.
* Real amplitude variational structure:
* Builds on RealAmplitudes design with five variational layers.
* Each layer implements alternating rotation blocks:
* Three-axis rotations (RX, RY, RZ) for single-qubit operations.
* Full entanglement pattern using CNOT gates between adjacent qubits.
* All parameters initialized randomly from normal distribution N(0, 1).
* Comprises (five layers \(\times\) qubits \(\times\) three rotations) trainable parameters.  
During the circuit creation, we ensure that circuits are created with variational layers limited to five layers and full entanglement scheme, producing a rich quantum feature space and leveraging superposition and entanglement for enhanced pattern recognition. The number of variational layers is limited to avoid issues such as the barren plateau problem [32], where the gradients of the cost function can shrink or vanish exponentially with an increase in the circuit depth or number of qubits. The randomly initialized parameters are optimized during training to learn the classification task while maintaining quantum mechanical properties throughout the computation, allowing the circuit to capture complex data patterns through the multi-layer quantum transformations. Figure 3 represents the quantum circuit. Initial Hadamard gates (H) are shown in the orange boxes at the start of each qubit line; the red boxes represent the parameterized rotation gates (RX, RY, RZ). Each rotation gate displays its initialized random parameter value (e.g., RY = -0.853, RX = -0.626), while the CNOT connections between adjacent qubits show the full entanglement pattern.