Towards Antisymmetric Neural Ansatz Separation
This work addresses a foundational problem in quantum chemistry for researchers modeling Fermionic systems, offering a novel theoretical separation that is incremental in advancing understanding of antisymmetric function representations.
The paper tackles the problem of separating two fundamental models of antisymmetric functions, which are crucial for modeling wavefunctions in quantum chemistry, by constructing a function that is efficiently expressible in Jastrow form but requires exponentially many terms to approximate with Slater determinants. This result provides the first explicit quantitative separation between these Ansätze, with the exponential dependence on N^2.
We study separations between two fundamental models (or \emph{Ansätze}) of antisymmetric functions, that is, functions $f$ of the form $f(x_{σ(1)}, \ldots, x_{σ(N)}) = \text{sign}(σ)f(x_1, \ldots, x_N)$, where $σ$ 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 Ansätze: 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 Ansätze.