Ground state-based quantum feature maps

TL;DR

This paper proposes a quantum data embedding method using ground states of parameterized Hamiltonians, analyzing its properties and limitations to advance quantum machine learning model development.

Contribution

It introduces a ground state-based quantum feature map, compares it with Fourier models, and analyzes its spectral properties and expressivity limitations.

Findings

Ground state embeddings have spectra with rapidly growing degrees.

Spectral degeneracies and structured coefficients limit model expressivity.

Results guide the development of high-capacity quantum models for machine learning.

Abstract

We introduce a quantum data embedding protocol based on the preparation of a ground state of a parameterized Hamiltonian. We analyze the corresponding quantum feature map, recasting it as an adiabatic state preparation procedure with Trotterized evolution. We compare the properties of underlying quantum models with ubiquitous Fourier-type quantum models, and show that ground state embeddings can be described effectively by a spectrum with degree that grows rapidly with the number of qubits, corresponding to a large model capacity. We observe that the spectrum contains massive frequency degeneracies, and the weighting coefficients for the modes are highly structured, thus limiting model expressivity. Our results provide a step towards understanding models based on quantum data, and contribute to fundamental knowledge needed for building efficient quantum machine learning (QML) protocols. As non-trivial embeddings are crucial for designing QML protocols that cannot be simulated classically, our findings guide the search for high-capacity quantum models that can largely outperform classical models.