  Characterising quantum processes is a key task in and constitutes a challenge
for the development of quantum technologies, especially at the noisy
intermediate scale of today's devices. One method for characterising processes
is randomised benchmarking, which is robust against state preparation and
measurement (SPAM) errors, and can be used to benchmark Clifford gates. A
complementing approach asks for full tomographic knowledge. Compressed sensing
techniques achieve full tomography of quantum channels essentially at optimal
resource efficiency. So far, guarantees for compressed sensing protocols rely
on unstructured random measurements and can not be applied to the data acquired
from randomised benchmarking experiments. It has been an open question whether
or not the favourable features of both worlds can be combined. In this work, we
give a positive answer to this question. For the important case of
characterising multi-qubit unitary gates, we provide a rigorously guaranteed
and practical reconstruction method that works with an essentially optimal
number of average gate fidelities measured respect to random Clifford
unitaries. Moreover, for general unital quantum channels we provide an explicit
expansion into a unitary 2-design, allowing for a practical and guaranteed
reconstruction also in that case. As a side result, we obtain a new statistical
interpretation of the unitarity -- a figure of merit that characterises the
coherence of a process. In our proofs we exploit recent representation
theoretic insights on the Clifford group, develop a version of Collins'
calculus with Weingarten functions for integration over the Clifford group, and
combine this with proof techniques from compressed sensing.
