  The temporal evolution of a quantum system can be characterized by quantum
process tomography, a complex task that consumes a number of physical resources
scaling exponentially with the number of subsystems. An alternative approach to
the full reconstruction of a quantum channel allows selecting which coefficient
from its matrix description to measure, and how accurately, reducing the amount
of resources to be polynomial. The possibility of implementing this method is
closely related to the possibility of building a complete set of mutually
unbiased bases (MUBs) whose existence is known only when the dimension of the
Hilbert space is the power of a prime number. However, an extension of the
method that uses tensor products of maximal sets of MUBs, has been introduced
recently. Here we explicitly describe how to implement this algorithm to
selectively and efficiently estimate any parameter characterizing a quantum
process in a non-prime power dimension, and we conducted for the first time an
experimental verification of the method in a Hilbert space of dimension $d=6$.
That is the small space for which there is no known a complete set of MUBs but
it can be decomposed as a tensor product of two other Hilbert spaces of
dimensions $D_1=2$ and $D_2=3$, for which a complete set of MUBs is known. The
$6$-dimensional states were codified in the discretized transverse momentum of
the photon wavefront. The state preparation and detection stages are
dynamically programmed with the use of only-phase spatial light modulators, in
a versatile experimental setup that allows to implement the algorithm in any
finite dimension.
