A minimal and universal representation of fermionic wavefunctions (fermions = bosons + one)

TL;DR

This work tackles the long-standing challenge of efficiently representing fermionic wavefunctions by converting antisymmetric functions into symmetric ones on an enlarged space through a minimal signature encoder $\boldsymbol{\eta}$. The authors establish a parity-graded representation where a symmetric function of augmented coordinates, $\Psi(\mathbf{R},\boldsymbol{\eta})$, encodes Fermi statistics with $c=1$ or $2$, and the original fermionic state is recovered via an odd projection involving $\boldsymbol{\eta}$. They provide explicit constructions for signature encoders in 1D, 2D, and periodic boundary conditions in higher dimensions, and formulate a universal representation in terms of a permutation-invariant feature map $\boldsymbol{\xi}$ and an antisymmetric $\boldsymbol{\eta}$, yielding $\psi(\mathbf{R})=\frac{f(\boldsymbol{\xi}(\mathbf{R}),\boldsymbol{\eta}(\mathbf{R}))-f(\boldsymbol{\xi}(\mathbf{R}),-\boldsymbol{\eta}(\mathbf{R}))}{2}$. The feature dimension scales as $D\sim N^d$ or $D\sim N$ depending on the embedding strategy, offering a more efficient alternative to previous antisymmetric-only representations. This framework provides a rigorous foundation for scalable, systematically improvable neural-network solvers for many-electron systems while preserving exact fermionic antisymmetry by construction.

Abstract

Representing fermionic wavefunctions efficiently is a central problem in quantum physics, chemistry and materials science. In this work, we introduce a universal and exact representation of continuous antisymmetric functions by lifting them to continuous symmetric functions defined on an enlarged space. Building on this lifting, we obtain a \emph{parity-graded representation} of fermionic wavefunctions, expressed in terms of symmetric feature variables that encode particle configuration and antisymmetric feature variables that encode exchange statistics. This representation is both exact and minimal: the number of required features scales as $D\sim N^d$ ($d$ is spatial dimension) or $D\sim N$ depending on the symmetric feature maps employed. Our results provide a rigorous mathematical foundation for efficient representations of fermionic wavefunctions and enable scalable and systematically improvable neural network solvers for many-electron systems.