Effect of data encoding on the expressive power of variational quantum-machine-learning models

TLDR

Quantum computers enable supervised learning by mapping data inputs to predictions via parametrized circuits, yet many theoretical properties of these models remain unexplored. The study examines how data‑encoding strategies affect the expressive power of parametrized quantum circuits as function approximators. Repeating simple data‑encoding gates allows quantum models to access richer frequency spectra. Quantum models can be expressed as partial Fourier series whose accessible frequencies depend on data‑encoding gates, and when the frequency spectrum is asymptotically rich, these models can realize any set of Fourier coefficients, making them universal function approximators.

Abstract

Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized 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 realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.