Waveflow: boundary-conditioned normalizing flows applied to fermionic wavefunctions

An efficient and expressive wavefunction ansatz is key to scalable solutions for complex many-body electronic structures. While Slater determinants are predominantly used for constructing antisymmetric electronic wavefunction ans\"{a}tze, this construction can result in limited expressiveness when the targeted wavefunction is highly complex. In this work, we introduce Waveflow, an innovative framework for learning many-body fermionic wavefunctions using boundary-conditioned normalizing flows. Instead of relying on Slater determinants, Waveflow imposes antisymmetry by defining the fundamental domain of the wavefunction and applying necessary boundary conditions. A key challenge in using normalizing flows for this purpose is addressing the topological mismatch between the prior and target distributions. We propose using O-spline priors and I-spline bijections to handle this mismatch, which allows for flexibility in the node number of the distribution while automatically maintaining its square-normalization property. We apply Waveflow to a one-dimensional many-electron system, where we variationally minimize the system's energy using variational quantum Monte Carlo (VQMC). Our experiments demonstrate that Waveflow can effectively resolve topological mismatches and faithfully learn the ground-state wavefunction.