Gaussian states play a central role in continuous quantum mechanics, and an appropriate generalization might play a similar role in discrete quantum mechanics. A common-sense discretization of these turns out to be trivial. In this work, we describe two new discretizations based on recent developments in discrete phase space methods. We also extend existing notions of convergence between continuous and discrete systems and apply some of them to our constructions. Accompanying this work is a numerical software package "qWeyl" which we use to give examples and provide visualizations.