Using Tensor Lattice Field Theory (for a recent review see arXiv:2010.06539),
we construct a gauge-invariant transfer matrix for compact scalar electrodynamics in arbitrary dimensions. We propose a noise-robust way to implement Gauss's law. We discuss quantum simulation experiments with Rydberg atoms where the electric field Hilbert space is approximated by a spin-1 triplet. We propose two different quantum simulators for this model obtained by assembling arrays of Rydberg atoms with ladder structures. We compare observables associated with the real-time evolution of (small) systems for the simulators and the target model. We briefly discuss recent experimental progress.