  A fundamental task in photonics is to characterise an unknown optical
process, defined by properties such as birefringence, spectral response,
thickness and flatness. Amongst many ways to achieve this, single-photon probes
can be used in a method called quantum process tomography (QPT). Furthermore,
QPT is an essential method in determining how a process acts on quantum
mechanical states. For example for quantum technology, QPT is used to
characterise multi-qubit processors and quantum communication channels; across
quantum physics QPT of some form is often the first experimental investigation
of a new physical process, as shown in the recent research into coherent
transport in biological mechanisms. However, the precision of QPT is limited by
the fact that measurements with single-particle probes are subject to
unavoidable shot noise---this holds for both single photon and laser probes. In
situations where measurement resources are limited, for example, where the
process is rapidly changing or the time bandwidth is constrained, it becomes
essential to overcome this precision limit. Here we devise and demonstrate a
scheme for tomography which exploits non-classical input states and quantum
interferences; unlike previous QPT methods our scheme capitalises upon the
possibility to use simultaneously multiple photons per mode. The
efficiency---quantified by precision per photon used---scales with larger
photon number input states. Our demonstration uses four-photon states and our
results show a substantial reduction of statistical fluctuations compared to
traditional QPT methods---in the ideal case one four-photon probe state yields
the same amount of statistical information as twelve single probe photons.
