  We propose a general scheme for a digital construction of lattice gauge
theories with dynamical fermions. In this method, the four-body interactions
arising in models with $2+1$ dimensions and higher, are obtained
stroboscopically, through a sequence of two-body interactions with ancillary
degrees of freedom. This yields stronger interactions than the ones obtained
through pertubative methods, as typically done in previous proposals, and
removes an important bottleneck in the road towards experimental realizations.
The scheme applies to generic gauge theories with Lie or finite symmetry
groups, both Abelian and non-Abelian. As a concrete example, we present the
construction of a digital quantum simulator for a $\mathbb{Z}_{3}$ lattice
gauge theory with dynamical fermionic matter in $2+1$ dimensions, using
ultracold atoms in optical lattices, involving three atomic species,
representing the matter, gauge and auxiliary degrees of freedom, that are
separated in three different layers. By moving the ancilla atoms with a proper
sequence of steps, we show how we can obtain the desired evolution in a clean,
controlled way.
