Quantum Machine Learning in Feature Hilbert Spaces

TLDR

Quantum computing shares a core idea with kernel methods: efficiently operating in an otherwise intractably large Hilbert space. The paper investigates this link and proposes two quantum approaches for classification, opening a new avenue for quantum machine learning algorithms. By treating quantum state encoding as a nonlinear feature map, the authors show that a quantum device can compute inner‑product kernels for classical kernel methods or use a variational circuit as a linear classifier directly in Hilbert space. They demonstrate the concepts with a squeezing‑based feature map in a continuous‑variable system and visualize its behavior on two‑dimensional benchmark datasets.

Abstract

A 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 Letter 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 analyze 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. The kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we 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 visualize the working principle with two-dimensional minibenchmark datasets.