  We propose and evaluate experimentally an approach to quantum process
tomography that completely removes the scaling problem plaguing the standard
approach. The key to this simplification is the incorporation of prior
knowledge of the class of physical interactions involved in generating the
dynamics, which reduces the problem to one of parameter estimation. This allows
part of the problem to be tackled using efficient convex methods, which, when
coupled with a constraint on some parameters allows globally optimal estimates
for the Kraus operators to be determined from experimental data. Parameterising
the maps provides further advantages: it allows the incorporation of mixed
states of the environment as well as some initial correlation between the
system and environment, both of which are common physical situations following
excitation of the system away from thermal equilibrium. Although the approach
is not universal, in cases where it is valid it returns a complete set of
positive maps for the dynamical evolution of a quantum system at all times.
