The effect of data encoding on the expressive power of variational quantum machine learning models
This work addresses a theoretical gap in quantum machine learning by analyzing the impact of data encoding on model expressiveness, which is incremental but provides foundational insights for the field.
The paper investigates how data encoding strategies affect the expressive power of variational quantum machine learning models, showing that these models can be expressed as partial Fourier series with frequencies determined by encoding gates, and that repeated encoding gates enable rich frequency spectra, leading to universal function approximation under certain conditions.
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.