  The basic idea of quantum computing is surprisingly similar to that of kernel
methods in machine learning, namely to efficiently perform computations in an
intractably large Hilbert space. In this paper we explore some theoretical
foundations of this link and show how it opens up a new avenue for the design
of quantum machine learning algorithms. We interpret the process of encoding
inputs in a quantum state as a nonlinear feature map that maps data to quantum
Hilbert space. A quantum computer can now analyse the input data in this
feature space. Based on this link, we discuss two approaches for building a
quantum model for classification. In the first approach, the quantum device
estimates inner products of quantum states to compute a classically intractable
kernel. This kernel can be fed into any classical kernel method such as a
support vector machine. In the second approach, we can use a variational
quantum circuit as a linear model that classifies data explicitly in Hilbert
space. We illustrate these ideas with a feature map based on squeezing in a
continuous-variable system, and visualise the working principle with
$2$-dimensional mini-benchmark datasets.
