Framework for efficient ab initio electronic structure with Gaussian Process States

We present a general framework for the efficient simulation of realistic fermionic systems with modern machine-learning-inspired representations of quantum many-body states, towards a universal tool for ab initio electronic structure. These machine-learning-inspired Ans\"atze have recently come to the fore in both a (first-quantized) continuum and discrete Fock space representations, where, however, the inherent scaling of the latter approach for realistic interactions has so far limited practical applications. With application to the ``Gaussian Process State,'' a recently introduced Ansatz inspired by systematically improvable kernel models in machine learning, we discuss different choices to define the representation of the computational Fock space. We show how local representations are particularly suited for stochastic sampling of expectation values, while also indicating a route to overcome the discrepancy in the scaling compared with continuum-formulated models. We are able to show competitive accuracy for systems with up to 64 electrons, including a simplified (yet fully ab initio) model of the Mott transition in three-dimensional hydrogen, indicating a significant improvement over similar approaches, even for moderate numbers of configurational samples.