  The simulation of lattice gauge theories with tensor network (TN) methods is
becoming increasingly fruitful. The vision is that such methods will,
eventually, be used to simulate theories in $(3+1)$ dimensions in regimes
difficult for other methods. So far, however, TN methods have mostly simulated
lattice gauge theories in $(1+1)$ dimensions. The aim of this paper is to
explore the simulation of quantum electrodynamics (QED) on infinite lattices
with TNs, i.e., fermionic matter fields coupled to a $U(1)$ gauge field,
directly in the thermodynamic limit. With this idea in mind we first consider a
gauge-invariant iDMRG simulation of the Schwinger model -i.e., QED in
$(1+1)d$-. After giving a precise description of the numerical method, we
benchmark our simulations by computing the substracted chiral condensate in the
continuum, in good agreement with other approaches. Our simulations of the
Schwinger model allow us to build intuition about how a simulation should
proceed in $(2+1)$ dimensions. Based on this, we propose a variational ansatz
using infinite Projected Entangled Pair States (PEPS) to describe the ground
state of $(2+1)d$ QED. The ansatz includes $U(1)$ gauge symmetry at the level
of the tensors, as well as fermionic (matter) and bosonic (gauge) degrees of
freedom both at the physical and virtual levels. We argue that all the
necessary ingredients for the simulation of $(2+1)d$ QED are, a priori, already
in place, paving the way for future upcoming results.
