Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces

TL;DR

This paper proves that quantum-enhanced feature spaces enable quantum machine learning models to universally approximate continuous functions, providing a theoretical foundation for their broad applicability.

Contribution

It establishes the universal approximation property of quantum feature map-based models, a key theoretical insight into their expressive power.

Findings

Quantum feature maps can approximate any continuous function.

Quantum models are capable of classifying disjoint regions.

Theoretical analysis supports broader application of quantum machine learning.

Abstract

Encoding classical data 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. The crucial idea is using the quantum Hilbert space as a quantum-enhanced feature space in machine learning models. 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 machine learning models induced from the quantum-enhanced feature space are universal approximators of continuous functions under typical quantum feature maps. We also study the capability of quantum feature maps in the classification of disjoint regions. Our work enables an important theoretical analysis to ensure that 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.