(1) and evaluating particular linear combinations thereof in a classical post-processing
step. The demerit of this approach is that the number of individual channel measure-
ments required to evaluate a single Pauli expectation value scales with the dimension of
the underlying Hilbert space. The measurement model studied here does not require such
a coarse-graining: every natural measurement (1) itself already corresponds to a valid
measurement instance. This considerably reduces the number of diﬀerent measurement
settings that are required in order to acquire suﬃcient data.
Randomized benchmarking has also been adapted to allow for process tomography
[18, 27, 28]. Ref.  adopted randomized benchmarking techniques to obtain a process
tomography protocol that is robust towards state preparation and measurement errors
(SPAM). This is a distinct advantage over other protocols that do not share this additional
feature. However, the trade of between an increase of the number of channel uses and the
gained robustness towards SPAM errors remains unclear.
Finally, Ref.  also considered process tomography via compressed sensing tech-
niques. However, this method is somewhat diﬀerent from the other approaches presented
here: Instead of assuming low Kraus rank, they consider processes that are element-wise
sparse with respect to a known basis. Prior knowledge of this sparsifying basis is a neces-
sary prerequisite for this approach. In turn, techniques from traditional compressed sensing
[29, 30] are applied (rather than low-rank matrix reconstruction protocols) to reconstruct
such processes from a number of measurements that is proportional to the sparsity and
depends only logarithmically on the ambient system sizes. While requiring only very few
measurement settings, the main disadvantage of this approach are stronger model assump-
tions (sparsity and knowledge of the sparsifying basis) and measurements that may be
challenging to implement in practice.
1.4 Experimental considerations
In several physically important platforms, process tomography has been experimentally
realized [26, 31, 32], both in the direct and hence ancilla-free [31, 32] and ancilla-based 
reading of the task. These works make use of diﬀerent preparations and measurements. It
should be clear, however, that in several physical architectures, random measurements of
the type discussed here can readily experimentally be implemented.
Speciﬁcally, Haar random unitary maps have been realized in a 16-dimensional Hilbert
space associated with the 6S
ground state of
Cs atoms . This has been achieved
by suitably exploiting a time-dependent Hamiltonian evolution giving rise to Haar random
unitaries, making use of methods of quantum control. Such random unitaries have been
put to use in a tomographic protocol, in an approach of performing quantum tomography
based on Haar random unitaries that builds upon earlier theoretical ideas laid out in Ref.
. The same idea of generating Haar random unitaries using suitable time-dependent
Hamiltonians should readily apply to systems of trapped ions , where a suitable type of
control can be achieved with present technology. Indeed, in a diﬀerent context, methods of
optimal control have readily been applied to optimize quantum gates for trapped ions .
Random unitaries, albeit not Haar distributed, that allow for quantum state tomogra-
phy of quantum many-body systems have been theoretically considered , making use
of operations only that can be considered basically feasible in experiments with cold atoms
in optical lattices , such as optical super-lattices and time-of-ﬂight measurements.
In integrated linear optical architectures [39, 40], Haar-random circuits are readily
conceivable . It is this type of setting in which the methods presented here are most
applicable, even though they apply to any physical system in which unitary 4-designs
Accepted in Quantum 2019-07-16, click title to verify. Published under CC-BY 4.0. 6