On the average-case complexity of learning states from the circular and Gaussian ensembles

Studying the complexity of states sampled from various ensembles is a central component of quantum information theory. In this work we establish the average-case hardness of learning, in the statistical query model, the Born distributions of states sampled uniformly from the compact symmetric spaces of type AI, AII, and DIII. These ensembles correspond in various senses to the so called circular and (fermionic) Gaussian ensembles. This finding complements analogous recent results for states sampled from the classical compact groups. On the technical side, we employ a somewhat unconventional approach to integrating over the compact groups which may be of some independent interest. For example, our approach allows us to exactly evaluate the total variation distances between the output distributions of Haar random unitary and orthogonal circuits and the constant distribution, which were previously known only approximately.