Multidimensional Fourier series with quantum circuits

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