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Variational minimization of tensor network states enables the exploration oflow energy states of lattice gauge theories. However, the exact numericalevaluation of high-dimensional tensor network states remains challenging ingeneral. In [E. Zohar, J. I. Cirac, Phys. Rev. D 97, 034510 (2018)] it wasshown how, by combining gauged Gaussian projected entangled pair states with avariational Monte Carlo procedure, it is possible to efficiently computephysical observables. In this paper we demonstrate how this approach can beused to investigate numerically the ground state of a lattice gauge theory.More concretely, we explicitly carry out the variational Monte Carlo procedurebased on such contraction methods for a pure gauge Kogut-Susskind Hamiltonianwith a $\mathbb{Z}_3$ gauge field in two spatial dimensions. This is a firstproof of principle to the method, which provides an inherent way to increasethe number of variational parameters and can be readily extended to systemswith physical fermions.

Variational Monte Carlo simulation with tensor networks of a pure $\mathbb{Z}_3$ gauge theory in (2+1)d