Towards Antisymmetric Neural Ansatz Separation

TL;DR

This paper demonstrates a fundamental separation between two key antisymmetric function models used in quantum chemistry, showing that some functions can be efficiently represented in one model but not the other without exponential complexity.

Contribution

The authors construct an explicit antisymmetric function that can be efficiently expressed with a Jastrow ansatz but cannot be approximated by Slater determinants without exponential complexity.

Findings

Jastrow ansatz can efficiently represent certain antisymmetric functions.

Slater determinants require exponential complexity to approximate some functions.

First explicit quantitative separation between Slater and Jastrow ans"atze.

Abstract

We study separations between two fundamental models (or \emph{Ans\"atze}) of antisymmetric functions, that is, functions $f$ of the form $f(x_{\sigma(1)}, \ldots, x_{\sigma(N)}) = \text{sign}(\sigma)f(x_1, \ldots, x_N)$, 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.