  Quantum process tomography is the task of reconstructing unknown quantum
channels from measured data. In this work, we introduce compressed
sensing-based methods that facilitate the reconstruction of quantum channels of
low Kraus rank. Our main contribution is the analysis of a natural measurement
model for this task: We assume that data is obtained by sending pure states
into the channel and measuring expectation values on the output. Neither
ancillary systems nor coherent operations across multiple channel uses are
required. Most previous results on compressed process reconstruction reduce the
problem to quantum state tomography on the channel's Choi matrix. While this
ansatz yields recovery guarantees from an essentially minimal number of
measurements, physical implementations of such schemes would typically involve
ancillary systems. A priori, it is unclear whether a measurement model tailored
directly to quantum process tomography might require more measurements. We
establish that this is not the case. Technically, we prove recovery guarantees
for three different reconstruction algorithms. The reconstructions are based on
a trace, diamond, and $\ell_2$-norm minimization, respectively. Our recovery
guarantees are uniform in the sense that with one random choice of measurement
settings all quantum channels can be recovered equally well. Moreover,
stability against arbitrary measurement noise and robustness against violations
of the low-rank assumption is guaranteed. Numerical studies demonstrate the
feasibility of the approach.
