Gauge theories are the basis of our understanding of nature, both as fundamental theories in particle physics and as emerging theories in condensed matter. Nowadays, our quantitative understanding of particle physics is based on the Langrangian approach. Perturbation theory has allowed for very precise predictions of the radiative corrections in the Standard model. Quantum Monte Carlo on the lattice have allowed for computing non-perturbative quantities like the spectrum of the Hadrons. Recently, classical and quantum simulation of Hamiltonian lattice gauge theories has emerged as a very promising route to access non-perturbative phenomena that are beyond the reach of Monte Carlo, especially due to the sign problem. So far, such approaches have been successful in one spatial dimension. The current challenge is to how to treat phenomenologically relevant gauge theories that live in higher dimensions, and in particular how to take the continuum limit in such simulations. In this talk I will tackle this challenge by taking quantum electrodynamics (QED) in 2+1 dimensions as a concrete example. After a brief introduction to the Hamiltonian approach and a review of theoretical and experimental progress, I will illustrate the two main challenges to simulate QED in a quantum simulator: i) how to physically realize magnetic (plaquette) interactions and ii) how to extract predictions in the continuum limit with finite resources. I will present concrete solutions that exploit the electro-magnetic duality and a proper regularization of the magnetic excitations. The latter strategy allows to efficiently simulate QED at arbitrary weak coupling and thus to access the renormalization of the coupling in near-term experiments also in the regimes where Markov chain Monte Carlo is hindered by an arbitrary increase of autocorrelation time.