Unifying view of fermionic neural network quantum states: From neural network backflow to hidden fermion determinant states

Among the variational wave functions for Fermionic Hamiltonians, neural network backflow (NNBF) and hidden fermion determinant states (HFDS) are two prominent classes to provide accurate approximations to the ground state. Here we develop a unifying view of fermionic neural quantum states casting them all in the framework of NNBF. NNBF wave-functions have configuration-dependent single-particle orbitals (SPO) which are parameterized by a neural network. We show that HFDS with $r$ hidden fermions can be written as a NNBF with an $r \times r$ determinant Jastrow and a restricted low-rank $r$ additive correction to the SPO. Furthermore, we show that in NNBF wave-functions, such determinant Jastrow's can generically be removed at the cost of further complicating the additive SPO correction increasing its rank by $r$. We numerically and analytically compare additive SPO corrections generated by the product of two matrices with inner dimension $r$. We find that larger $r$ wave-functions span a larger space and give evidence that simpler and more direct updates to the SPO's tend to be more expressive and better energetically. These suggest the standard NNBF approach is preferred amongst other related choices. Finally, we uncover that the row-selection used to select single-particle orbitals allows significant sign and amplitude modulation between nearby configurations and is partially responsible for the quality of NNBF and HFDS wave-functions.