O(N2) Universal Antisymmetry in Fermionic Neural Networks

Fermionic neural network (FermiNet) is a recently proposed wavefunction Ansatz, which is used in variational Monte Carlo (VMC) methods to solve the many-electron Schr"odinger equation. FermiNet proposes permutation-equivariant architectures, on which a Slater determinant is applied to induce antisymmetry. FermiNet is proved to have universal approximation capability with a single determinant, namely, it suffices to represent any antisymmetric function given sufficient parameters. However, the asymptotic computational bottleneck comes from the Slater determinant, which scales with O(N3) for N electrons. In this paper, we substitute the Slater determinant with a pairwise antisymmetry construction, which is easy to implement and can reduce the computational cost to O(N2). We formally prove that the pairwise construction built upon permutation-equivariant architectures can universally represent any antisymmetric function. Besides, this universality can be achieved via continuous approximators when we aim to represent ground-state wavefunctions.