  An optical waveplate rotating light polarization can be modeled as a
single-qubit unitary operator, whose action can be experimentally determined
via quantum process tomography. Standard approaches to tomographic problems
rely on the maximum-likelihood estimation, providing the most likely
transformation to yield the same outcomes as a set of experimental projective
measurements. The performances of this method strongly depend on the number of
input measurements and the numerical minimization routine that is adopted. Here
we investigate the application of genetic and machine learning approaches to
this problem, finding that both allow for accurate reconstructions and fast
operations when processing a set of projective measurements very close to the
minimal one. We apply these techniques to the case of space-dependent
polarization transformations, providing an experimental characterization of the
optical action of spin-orbit metasurfaces having patterned birefringence. Our
efforts thus expand the toolbox of methodologies for optical process
tomography. In particular, we find that the neural network-based scheme
provides a significant speed-up, that may be critical in applications requiring
a characterization in real-time. We expect these results to lay the groundwork
for the optimization of tomographic approaches in more general quantum
processes, including non-unitary gates and operations in higher-dimensional
Hilbert spaces.
