Concentration of Data Encoding in Parameterized Quantum Circuits

Variational quantum algorithms have been acknowledged as the leading strategy to realize near-term quantum advantages in meaningful tasks, including machine learning and optimization. When applied to tasks involving classical data, such algorithms generally begin with data encoding circuits and train quantum neural networks (QNNs) to minimize target functions. Although QNNs have been widely studied to improve these algorithms' performance on practical tasks, there is a gap in systematically understanding the influence of data encoding on the eventual performance. In this paper, we make progress in filling this gap by considering the common data encoding strategies based on parameterized quantum circuits. We prove that, under reasonable assumptions, the distance between the average encoded state and the maximally mixed state could be explicitly upper-bounded with respect to the width and depth of the encoding circuit.