Geometry of backflow transformation ansatz for quantum many-body fermionic wavefunctions

TL;DR

This paper investigates the geometric limitations of backflow transformation ansatz in representing antisymmetric wavefunctions, showing it requires exponentially many determinants for general cases, thus revealing fundamental constraints.

Contribution

It provides a geometric analysis demonstrating that backflow ansatz cannot efficiently represent all antisymmetric polynomials, requiring exponentially many determinants.

Findings

Backflow ansatz cannot efficiently represent all antisymmetric polynomials.

Representing a generic antisymmetric polynomial needs exponentially many determinants.

The study bounds the dimensions of the ansatz's source and target spaces.

Abstract

Wave function ansatz based on the backflow transformation are widely used to parametrize anti-symmetric multivariable functions for many-body quantum problems. We study the geometric aspects of such ansatz, in particular we show that in general totally antisymmetric polynomials cannot be efficiently represented by backflow transformation ansatz at least in the category of polynomials. In fact, one needs a linear combination of at least $O(N^{3N-3})$ determinants to represent a generic totally antisymmetric polynomial. Our proof is based on bounding the dimension of the source of the ansatz from above and bounding the dimension of the target from below.