# Hybrid Quantum Neural Network Image Anti-Noise Classification Model Combined with Error Mitigation
## 3 Construction of Mixed Variational Quantum Neural Network and Error Mitigation Model
### Construction of QNN by Variational Quantum Circuit  
In this research endeavor, we delve into the realm of quantum machine learning, specifically exploring the integration of amplitude encoding within QNN for image classification. Our methodology draws inspiration from the QNN model proposed by [PERSON] et al. [41], where VQC is leveraged for constructing the QNN. This novel approach seeks to harness the quantum advantage in processing and representing image features through the amplitude encoding scheme. Quantum neural networks operate on quantum input data through a sequence of parameter-dependent quantum gates. The representation of a QNN can be expressed as:  
\[U(\theta)=\prod_{c=1}^{N}V_{c}U_{c}(\theta_{c}) \tag{10}\]  
Here, \(\theta\) represents the trainable parameters, \(V_{c}\) and \(U_{c}(\theta_{c})\) are quantum gates acting on the data, forming a powerful quantum circuit capable of complex transformations. Given an input, \(x\), and trainable parameters \(\theta\), the amplitude encoding scheme transforms the input into a quantum state  
\[|\Psi\rangle=U_{\theta}|x\rangle \tag{11}\]  
For binary classification, observable projections \(O_{k}^{+}andO_{k}^{-}\) on the PauliZ basis are defined, corresponding to spin +1 and \(-1\), respectively. \[P_{1}(|\Psi\rangle)=\langle\Psi|O_{k}^{+}|\Psi\rangle \tag{12}\]  
Similarly, \(P_{2}(|\Psi_{i}\rangle)\) is defined for Class 2. Classification for a new input \(|\Psi_{i}\rangle\) is determined based on the product of probabilities, with the class receiving the highest probability being assigned as the prediction. The output of the QNN is based on the measurement results of quantum bits. The expectation value E is used as the QNN output, defined as:  
\[E=\langle\Psi_{x}|U^{\mathrm{T}}(\theta)VU(\theta)|\Psi_{x}\rangle \tag{13}\]  
Here, \(V\) is a linear combination of Pauli operators applied to the observable of the reading quantum bit, providing a probabilistic representation of the quantum state. The construction of the quantum classifier involves a variational quantum circuit utilizing \(s\) data quantum bits. These bits load local features extracted through CAE and  
Figure 3: The figure show s an example of bottom–top amplitude coding. The black dots in the circuit represent the control relationship between qubits, and the crosses represent the interaction or coupling between two or more qubits. Where the quantum logic gate corresponding to the circuit in the figure is the controlled SWAP gate, which is used to exchange auxiliary bits and output bit quantum states. amplitude encoding. The prediction is obtained through a single quantum bit output, with the number of reading quantum bits determined by the classification task. In our approach, the Ising coupling gate entangles the qubits. Similar to classical neural networks, the variable component subgate functions as a neuron, and the block serves as a layer in QNN. A block is defined as m gates continuously acting on m pairs of data and readout qubits. This block can be repeated to construct a complex QNN with more parameters. The prediction layer for the data is denoted as \(U_{p}(\theta_{p})\), and the predicted result for a given instance \(x\) is represented as:  
\[y^{\prime}=\langle\Psi_{T}|U_{p}^{\dagger}(\theta_{p})VU_{p}(\theta_{p})|\Psi_ {T}\rangle \tag{14}\]  
where \(|\Psi_{T}\rangle\) is the quantum state \(T\) whose extracted features are encoded by amplitude. The circuit is shown in Figure 4. The prediction layer utilizes a VQC that employs double-qubit parametrized gates, inducing entanglement between quantum bits. The optimization strategy follows the scheme proposed by [PERSON] et al. [42], incorporating the cross-entropy as the chosen loss function. The loss function is defined as the negative log-likelihood of the predicted probabilities compared to the true labels.