  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.
