  Gate set tomography (GST) allows for a self-consistent characterization of
noisy quantum information processors. The standard device-agnostic approach
treats the QIPs as black boxes that are only constrained by the laws of
physics, attaining full generality at a considerable resource cost: numerous
circuits built from the gate set must be run in order to amplify each of the
gate set parameters. In this work, we show that a microscopic parametrization
of quantum gates under time-correlated noise on the driving phase, motivated by
recent experiments with trapped-ion gates, reduces the required resources
enabling a more efficient version of GST. By making use of the formalism of
filter functions over the noise spectral densities, we discuss the minimal
parametrizations of the gate set that include the effect of finite correlation
times and non-Markovian quantum evolutions during the individual gates. We
compare the estimated gate sets obtained by our method and the standard
long-sequence GST, discussing their accuracies in terms of established metrics,
as well as showcasing the advantages of the parametrized approach in terms of
the sampling complexity for specific examples.
