  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 analytically 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.
