  The solution of gauge theories is one of the most promising applications of
quantum technologies. Here, we discuss the approach to the continuum limit for
$U(1)$ gauge theories regularized via finite-dimensional Hilbert spaces of
quantum spin-$S$ operators, known as quantum link models. For quantum
electrodynamics (QED) in one spatial dimension, we numerically demonstrate the
continuum limit by extrapolating the ground state energy, the scalar, and the
vector meson masses to large spin lengths $S$, large volume $N$, and vanishing
lattice spacing $a$. By exactly solving Gauss' law for arbitrary $S$, we obtain
a generalized PXP spin model and count the physical Hilbert space dimension
analytically. This allows us to quantify the required resources for reliable
extrapolations to the continuum limit on quantum devices. We use a functional
integral approach to relate the model with large values of half-integer spins
to the physics at topological angle $\Theta=\pi$. Our findings indicate that
quantum devices will in the foreseeable future be able to quantitatively probe
the QED regime with quantum link models.
