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arXiv:2111.10314

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

Huang, Hang, Joseph M. Landsberg, Jianfeng Lu

Published Nov 19, 2021• 0 citations
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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.

DOI
10.48550/arxiv.2111.10314

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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(N3N−3)O(N^{3N-3})O(N3N−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.
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The basic question is addressed of how the space dimension ddd is encoded in the Hilbert space of NNN identical fermions. There appears a finite number $N{!}^{d\ensuremath{-}1}$ of many-body wave functions, called shapes, which cannot be generated by trivial combinatorial extension of the one-dimensional ones. A general algorithm is given to list them all in terms of standard Slater determinants. Conversely, excitations which can be induced from the one-dimensional case are bosonized into a system of distinguishable bosons, called Euler bosons, much like the electromagnetic field is quantized in terms of photons distinguishable by their wave numbers. Their wave functions are given explicitly in terms of elementary symmetric functions, reflecting the fact that the fermion sign problem is trivial in one dimension. The shapes act as vacua for the Euler bosons. They are the natural generalization of the single-Slater-determinant form for the ground state to more than one dimension. In terms of algebraic invariant theory, the shapes are antisymmetric invariants which finitely generate the NNN-fermion Hilbert space as a graded algebra over the ring of symmetric polynomials. Analogous results hold for identical bosons.
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O~(N2)\widetilde{O}(N^2)O(N2) Representation of General Continuous Anti-symmetric Function
In quantum mechanics, the wave function of fermion systems such as many-body electron systems are anti-symmetric (AS) and continuous, and it is crucial yet challenging to find an ansatz to represent them. This paper addresses this challenge by presenting an O~(N2){\widetilde O}(N^2)O(N2) ansatz based on permutation-equivariant functions. We prove that our ansatz can represent any AS continuous functions, and can accommodate the determinant-based structure proposed by Hutter [14], solving the proposed open problems that O(N){O}(N)O(N) Slater determinants are sufficient to provide universal representation of AS continuous functions. Together, we offer a generalizable and efficient approach to representing AS continuous functions, shedding light on designing neural networks to learn wave functions.