  We study quantum process tomography given the prior information that the map
is a unitary or close to a unitary process. We show that a unitary map on a
$d$-level system is completely characterized by a minimal set of $d^2{+}d$
elements associated with a collection of POVMs, in contrast to the $d^4{-}d^2$
elements required for a general completely positive trace-preserving map. To
achieve this lower bound, one must probe the map with a particular set of $d$
pure states. We further compare the performance of different compressed sensing
algorithms used to reconstruct a near-unitary process from such data. We find
that when we have accurate prior information, an appropriate compressed sensing
method reduces the required data needed for high-fidelity estimation, and
different estimators applied to the same data are sensitive to different types
of noise. Compressed sensing techniques can therefore be used both as
indicators of error models and to validate the use of the prior assumptions.
