Statistical learning on randomized data to verify quantum state k-designs

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.