  Quantum machine learning is the field that aims to integrate machine learning
with quantum computation. In recent years, the field has emerged as an active
research area with the potential to bring new insights to classical machine
learning problems. One of the challenges in the field is to explore the
expressibility of parametrized quantum circuits and their ability to be
universal function approximators, as classical neural networks are. Recent
works have shown that with a quantum supervised learning model, we can fit any
one-dimensional Fourier series, proving their universality. However, models for
multidimensional functions have not been explored in the same level of detail.
In this work, we study the expressibility of various types of circuit ansatzes
that generate multidimensional Fourier series. We found that, for some
ansatzes, the degrees of freedom required for fitting such functions grow
faster than the available degrees in the Hilbert space generated by the
circuits. For example, single-qudit models have limited power to represent
arbitrary multidimensional Fourier series. Despite this, we show that we can
enlarge the Hilbert space of the circuit by using more qudits or higher local
dimensions to meet the degrees of freedom requirements, thus ensuring the
universality of the models.
