Universal Approximation Property of Quantum Feature Map

Encoding classical inputs into quantum states is considered a quantum feature map to map classical data into a quantum Hilbert space. This feature map provides opportunities to incorporate quantum advantages into machine learning algorithms to be performed on near-term intermediate-scale quantum computers. While the quantum feature map has demonstrated its capability when combined with linear classification models in some specific applications, its expressive power from the theoretical perspective remains unknown. We prove that the quantum feature map is a universal approximator of continuous functions under its typical settings in many practical applications. We also study the capability of the quantum feature map in the classification of disjoint regions. Our work enables an important theoretical analysis to ensure that quantum-enhanced machine learning algorithms based on quantum feature maps can handle a broad class of machine learning tasks. In light of this, one can design a quantum machine learning model with more powerful expressivity.