  Learning an unknown quantum process is a central task for validation of the
functioning of near-term devices. The task is generally hard, requiring
exponentially many measurements if no prior assumptions are made on the
process. However, an interesting feature of the classically-simulable Clifford
group is that unknown Clifford operations may be efficiently determined from a
black-box implementation. We extend this result to the important class of
fermionic Gaussian operations. These operations have received much attention
due to their close links to fermionic linear optics. We then introduce an
infinite family of unitary gates, called the Matchgate Hierarchy, with a
similar structure to the Clifford Hierarchy. We show that the Clifford
Hierarchy is contained within the Matchgate Hierarchy and how operations at any
level of the hierarchy can be efficiently learned.
