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Fermionic neural network (FermiNet) is a recently proposed wavefunctionAnsatz, which is used in variational Monte Carlo (VMC) methods to solve themany-electron Schr\"{o}dinger equation. FermiNet proposespermutation-equivariant architectures, on which a Slater determinant is appliedto induce antisymmetry. FermiNet is proved to have universal approximationcapability with a single determinant, namely, it suffices to represent anyantisymmetric function given sufficient parameters. However, the asymptoticcomputational bottleneck comes from the Slater determinant, which scales with$O(N^3)$ for $N$ electrons. In this paper, we substitute the Slater determinantwith a pairwise antisymmetry construction, which is easy to implement and canreduce the computational cost to $O(N^2)$. We formally prove that the pairwiseconstruction built upon permutation-equivariant architectures can universallyrepresent any antisymmetric function. Besides, this universality can beachieved via continuous approximators when we aim to represent ground-statewavefunctions.

$O(N^2)$ Universal Antisymmetry in Fermionic Neural Networks