Gauging a superposition of fermionic Gaussian projected entangled pair states to get lattice gauge theory eigenstates

Gauged fermionic projected entangled pair states (GFPEPS) and their Gaussian counterpart (GGFPEPS) are a novel type of lattice gauge theory Ansatz state that combine ideas from the Monte Carlo and tensor network communities. In particular, computation of observables for such states boils down to a Monte Carlo integration over possible gauge field configurations that have probabilities dictated by a fermionic tensor network contraction that accounts for the matter in that background configuration. Crucially, this probability distribution is positive definite and real so that there is no sign problem. When the underlying PEPS is Gaussian, tensor network contraction can be done efficiently, and in this scenario the Ansatz has been tested well numerically. In this work we propose to gauge superpositions of Gaussian PEPS and demonstrate that one can still efficiently compute observables when few Gaussians are in the superposition. As we will argue, the latter is exactly the case for bound states on top of the strongly interacting LGT vacuum, which makes this Ansatz particularly suitable for that scenario. As a corollary, we will provide an exact representation of the LGT ground state as a gauged PEPS.