# 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
### Amplitude Coding  
In this study, we explore the utilization of amplitude encoding as an alternative to the encoding layer within QNN for image feature representation. The primary objective is to alleviate the complexity associated with traditional classification models. Moreover, employing amplitude coding in advance holds the potential to mitigate noise interference and bolster the robustness of resulting quantum states. This proactive approach contributes to the improvement of accuracy and stability in image classification tasks.  
Amplitude encoding serves as a powerful technique for mapping classical data onto quantum states [40]. Through this process, classical image data undergo a transformation, becoming embedded within the unique characteristics of quantum states. The intrinsic ability of quantum states to handle complex coherence and entanglement positions amplitude encoding as a potentially superior method for certain image processing tasks, particularly those requiring intricate feature extraction and classification.  
A notable advantage of early amplitude encoding lies in its capability to mitigate noise interference. The resulting quantum states exhibit enhanced robustness, a critical factor in the context of image classification. The reduction in noise interference contributes to an overall improvement in accuracy and stability, making amplitude encoding a valuable preprocessing step. For a given dataset, \(D^{\prime}=\{t^{j}\}_{j=1}^{\prime}\), \(t^{j}=\left(t_{1}^{j},t_{2}^{j},\cdots,t_{n}^{j}\right)\), \(l^{\prime}\) represents the total number of samples in the dataset, and \(n\) represents the number of features per sample. the representation function is defined as  
\[f\big{(}D^{\prime}\big{)}=\frac{1}{C}\sum_{j=1}^{l^{\prime}}\sum_{i=1}^{n}t_{i }^{i}|i,j\rangle \tag{8}\]  
where \(C\) is the normalization constant. This representation provides a structured basis for further exploration and analysis. Efficiency in amplitude encoding is highlighted by the fact that, for datasets with a length \(\ln\) that is a power of 2, only \(log(ln)\) quantum bits are required to encode \(ln\) amplitudes. The amplitude coding problem can be simplified to give a single-point set \(T=\{t=(t_{0},\cdots,t_{N-1})\}\subset\mathcal{R}^{N}\). Neglecting the normalization constant  
\[f(T)=\sum t_{i}|i\rangle \tag{9}\]  
To optimize runtime considerations, the introduction of N quantum bits allows for an expanded choice of amplitude encoding bases. The selection of the \(W\)-state basis exemplifies the strategic choices made in our approach.  
Here, we use bottom-top amplitude coding. The adoption of bottom-top amplitude encoding involves the construction of a quantum circuit with \(O(n)\) width and \(O(\lceil\log_{2}N\rceil)\) depth. Notably, the structure of the angle tree, specifically the leftmost subtree \((\alpha_{0},\alpha_{1},\alpha_{3})\), corresponds to the output qubit, with others serving as auxiliary qubits. The construction is meticulously illustrated in accompanying diagrams. The build form is shown in the following Figure 3.  
In conclusion, our investigation into amplitude encoding as a preprocessing step for image feature representation within a quantum framework highlights its potential to enhance classification models. The decrease in complexity, reduction in noise interference, and the enhanced robustness of quantum states collectively contribute to a more accurate and stable image classification process. The strategic considerations in amplitude encoding efficiency and the structured circuit construction further underscore the viability of this approach in the realm of quantum image processing.