  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 ${\widetilde O}(N^2)$ 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 ${O}(N)$
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.
