  Driven by the significant advantages offered by quantum computing, research
in quantum machine learning has increased in recent years. While quantum
speed-up has been demonstrated in some applications of quantum machine
learning, a comprehensive understanding of its underlying mechanisms for
improved performance remains elusive. Our study fills this gap by examining the
expressibility of quantum circuits integrated within a convolutional neural
network (CNN). Through numerical training on the MNIST dataset, our hybrid
quantum-classical CNN model exhibited superior feature selection capabilities
and significantly reduced the required training steps compared to the classical
CNN. To understand the root of this enhanced performance, we conducted an
analytical investigation of the functional expressibility of quantum circuits
and derived a quantum activation function. We demonstrated that this quantum
activation is more efficient in selecting important features and discarding
unimportant information of input images. These findings not only deepen our
comprehension of quantum-enhanced machine-learning models but also advance the
classical machine-learning technique by introducing the quantum-inspired
activation function.
