Quantum machine learning in feature Hilbert spaces

TL;DR

This paper explores the theoretical connection between quantum computing and kernel methods in machine learning, proposing quantum algorithms for classification that leverage Hilbert space representations.

Contribution

It introduces a framework linking quantum encoding to nonlinear feature maps and presents two quantum classification approaches using kernel estimation and variational circuits.

Findings

Quantum encoding acts as a nonlinear feature map.

Two quantum classification methods are proposed.

Illustrated with continuous-variable system examples.

Abstract

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.