Towards Antisymmetric Neural Ansatz Separation

We study separations between two fundamental models (or \emph{Ans\"atze}) of antisymmetric functions, that is, functions $f$ of the form $f(x{\sigma(1)}, \ldots, x{\sigma(N)}) = \text{sign}(\sigma)f(x1, \ldots, xN)$, where $\sigma$ is any permutation. These arise in the context of quantum chemistry, and are the basic modeling tool for wavefunctions of Fermionic systems. Specifically, we consider two popular antisymmetric Ans\"atze: the Slater representation, which leverages the alternating structure of determinants, and the Jastrow ansatz, which augments Slater determinants with a product by an arbitrary symmetric function. We construct an antisymmetric function in $N$ dimensions that can be efficiently expressed in Jastrow form, yet provably cannot be approximated by Slater determinants unless there are exponentially (in $N^2$) many terms. This represents the first explicit quantitative separation between these two Ans\"atze.