Universal Approximation Property of Quantum Feature Map

Encoding classical inputs into quantum states is considered as a quantum feature map to map the classical data into the quantum Hilbert space. This feature map paves opportunities to merge the advantages of quantum mechanics into machine learning algorithms to perform on the near-term intermediate-scale quantum computers. While the quantum feature map has demonstrated its capability when combining 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 further study the capability of the quantum feature map in the classification of disjoint regions. Our work enables a theoretical analysis of the feasibility of quantum-enhanced machine learning algorithms. In light of this, one can utilize knowledge to design a quantum machine learning model with more powerful expressivity.