  Quantum tomography is an essential method of the photonic technology toolbox
and is routinely used for evaluation of experimentally prepared states of light
and characterization of devices transforming such states. The tomography
procedure consists of many different sequentially performed measurements. We
present considerable tomography speedup by optimally arranging the individual
constituent measurements, which is equivalent to solving an instance of the
traveling salesman problem. As an example, we obtain solutions for photonic
systems of up to five qubits, and conclude that already for systems of three or
more qubits, the total duration of the tomography procedure can be halved. The
reported speedup has been verified experimentally for quantum state tomography
and also for full quantum process characterization up to six qubits, without
resorting to any complexity reduction or simplification of the system of
interest. Our approach is versatile and reduces the time of an input-output
characterization of optical devices and various scattering processes as well.
