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.

本文证明典型情况下，基于量子增强特征空间的机器学习模型是连续函数的通用逼近器，从理论角度探究了量子特征映射的表达能力与在分类不相交区域方面的功能。

量子增强特征空间中量子机器学习模型的普适逼近特性

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

Quantum classifiers are trainable quantum circuits used as machine learning
models. The first part of the circuit implements a quantum feature map that
encodes classical inputs into quantum states, embedding the data in a
high-dimensional Hilbert space; the second part of the circuit executes a
quantum measurement interpreted as the output of the model. Usually, the
measurement is trained to distinguish quantum-embedded data. We propose to
instead train the first part of the circuit -- the embedding -- with the
objective of maximally separating data classes in Hilbert space, a strategy we
call quantum metric learning. As a result, the measurement minimizing a linear
classification loss is already known and depends on the metric used: for
embeddings separating data using the l1 or trace distance, this is the Helstrom
measurement, while for the l2 or Hilbert-Schmidt distance, it is a simple
overlap measurement. This approach provides a powerful analytic framework for
quantum machine learning and eliminates a major component in current models,
freeing up more precious resources to best leverage the capabilities of
near-term quantum information processors.

该论文提出了一种称为量子度量学习的新策略，通过训练可训练的量子电路的第一部分来实现数据类最大分离的嵌入，从而消除了当前模型中的一个主要部分。