Communications in Mathematical Sciences

Volume 21 (2023)

Number 5

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

Pages: 1447 – 1453

(Fast Communication)



Hang Huang (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Joseph M. Landsberg (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Jianfeng Lu (Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, North Carolina, U.S.A.)


Wave function ansatze 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 ansatze, in particular we show that in general totally antisymmetric polynomials cannot be efficiently represented by backflow transformation ansatze at least in the category of polynomials. In fact, if there are $N$ particles in the system, 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 ansatze from above and bounding the dimension of the target from below.


backflow transformation, fermionic wavefunction, anti-symmetry, secant variety

2010 Mathematics Subject Classification

14Nxx, 70G75, 81-xx

Landsberg was supported by NSF grant AF-1814254. Lu was supported in part by NSF grants DMS-2012286 and CHE-2737263. This project is an outcome of the IPAM program: “Tensor Methods and Emerging Applications to the Physical and Data Sciences”, March 8, 2021 – June 11, 2021.

Received 27 May 2022

Received revised 23 August 2022

Accepted 9 October 2022

Published 30 August 2023