The effect of data encoding on the expressive power of variational quantum machine learning models

TL;DR

This paper explores how data encoding strategies affect the expressive power of variational quantum machine learning models, revealing that richer frequency spectra enable universal function approximation.

Contribution

It demonstrates that data encoding methods determine the frequency spectrum accessible to quantum models, impacting their ability to approximate functions.

Findings

Repeated encoding gates increase frequency spectrum richness.

Quantum models can realize all Fourier coefficients with sufficient encoding.

Asymptotically rich frequency spectra lead to universal approximation.

Abstract

Quantum computers can be used for supervised learning by treating parametrised quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate practical implications of this approach, many important theoretical properties of these models remain unknown. Here we investigate how the strategy with which data is encoded into the model influences the expressive power of parametrised quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data encoding gates in the circuit. By repeating simple data encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realise all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.