Free Fermion Distributions Are Hard to Learn

Free fermions are some of the best studied quantum systems. However, little is known about the complexity of learning free-fermion distributions. In this work we establish the hardness of this task in the particle number non-preserving case. In particular, we give an information theoretical hardness result for the general task of learning from expectation values and, in the more general case when the algorithm is given access to samples, we give a computational hardness result based on the LPN assumption for learning the probability density function.