  Random ensembles of pure states have proven to be extremely important in
various aspects of quantum physics such as benchmarking the performance of
quantum circuits, testing for quantum advantage, providing novel insights for
many-body thermalization and studying the black hole information paradox.
Although generating a fully random ensemble is experimentally challenging,
approximations of it are just as useful and are known to emerge naturally in a
variety of physical models, including Rydberg setups. These are referred to as
approximate quantum state designs, and verifying their degree of randomness can
be an expensive task, similar to performing full quantum state tomography on
many-body systems. In this theoretical work, we efficiently validate the
character of approximate quantum designs with respect to data size acquisition
when compared to the conventional frequentist approach. This is achieved by
translating the information residing in the complex many-body state into a
succinct representation of classical data using a random projective measurement
basis, which is then processed using methods of statistical inference such as
maximum likelihood estimation and neural networks and benchmarked against the
predictions of shadow tomography. Our scheme of combining machine learning
methods for postprocessing the data obtained from randomized measurements for
efficient characterisation of (approximate) quantum state k designs is
applicable to any noisy quantum platform that can generate quantum designs.
