  In this paper, we examine a compact $U(1)$ lattice gauge theory in $(2+1)$
dimensions and present a strategy for studying the running coupling and
extracting the non-perturbative $\Lambda$-parameter. To this end, we combine
Monte Carlo simulations and quantum computing, where the former can be used to
determine the numerical value of the lattice spacing $a$, and the latter allows
for reaching the perturbative regime at very small values of the bare coupling
and, correspondingly, small values of $a$. The methodology involves a series of
sequential steps (i.e., the step scaling function) to bridge results from small
lattice spacings to non-perturbative large-scale lattice calculations. Focusing
on the pure gauge case, we demonstrate that these quantum circuits, adapted to
gauge degrees of freedom, are able to capture the relevant physics by studying
the expectation value of the plaquette operator, for matching with
corresponding Monte Carlo simulations. We also present results for the static
potential and static force, which can be related to the renormalized coupling.
The procedure outlined in this work can be extended to Abelian and non-Abelian
lattice gauge theories with matter fields and might provide a way towards
studying lattice quantum chromodynamics utilizing both quantum and classical
methods.
