We show that the problem of classifying, up to isomorphism, the collection of zero-entropy mixing automorphisms of a probability space, is intractable. More precisely, the collection of isomorphic pairs of automorphisms in this class is not Borel, when considered as a subset of the Cartesian product of the collection of measure-preserving automorphisms with itself. This remains true if we restrict to zero-entropy mixing automorphisms that are also C∞ diffeomorphisms of the five-dimensional torus. In addition, both of these results still hold if “isomorphism” is replaced by “Kakutani equivalence.” This is joint work with Philipp Kunde.