  The characterization of a quantum device is a crucial step in the development
of quantum experiments. This is accomplished via Quantum Process Tomography,
which combines the outcomes of different projective measurements to deliver a
possible reconstruction of the underlying process. The tomography is typically
performed by processing an overcomplete set of measurements and extracting the
process matrix from maximum-likelihood estimation. Here, we introduce a new
technique, referred to as Fourier Quantum Process Tomography, which requires a
reduced number of measurements, and benchmark its performance against the
standard maximum-likelihood approach. Fourier Quantum Process Tomography is
based on measuring probability distributions in two conjugate spaces for
different state preparations and projections. Exploiting the concept of phase
retrieval, our scheme achieves a complete and robust characterization of the
setup by processing a near-minimal set of measurements. We experimentally test
the technique on different space-dependent polarization transformations,
reporting average fidelities higher than 90% and significant computational
advantage.
