The Effect of Data Encoding on Quantum Convolutional Neural Networks

Quantum Convolutional Neural Networks (QCNNs) are hybrid solutions suggested by literature to achieve good generalization with less data. In this work, we investigate the effect of the data encoding choice on the QCNN architecture, where the classical convolutional layer is replaced by a variational quantum circuit (VQC). The task is binary classification of malign or benign ultrasound images of a medical dataset (BreastMNIST). We first investigate quantum metrics from literature such as normalized effective dimension, entanglement capability and expressibility, but find no clear correlation with performance. We also explore the idea of VQCs as Fourier series. The QCNN architecture may consist of multiple reuploading layers each consisting of data encoding and a trainable Ansatz. We perform numerical experiments by varying the data encoding to be the angle or higher order encoding up to four reuploading layers. We analyze the distribution of the Fourier coefficients in each case and find evidence that reinforces ideas from literature: the variance of the distribution increases with the number of layers and appears to reach a saturation at two layers in the univariate case, which indicates that the circuits do not have enough degrees of freedom to allow full control over the Fourier coefficients. This can be interpreted as noise induced in the circuit due to lack of degrees of freedom. A better understanding of encoding strategies is needed to design an alternative strategy that is resilient to this issue. The multivariate case, where different input combinations are considered is analyzed for up to six data reuploading layers. The same observation of increased variance persists, and a saturation is reached at four layers. We find that the best performing models exhibit a linear structure in the distribution of coefficients, corresponding to a fixed phase. This might be an indication that a fixed phase simplifies the problem’s optimization and requires further investigation.