We perform a digital quantum simulation of a gauge theory with a topological
term in Minkowski spacetime, which is practically inaccessible by standard
lattice Monte Carlo simulations. We focus on $1+1$ dimensional quantum
electrodynamics with the $\theta$-term known as the Schwinger model. We
construct the true vacuum state of a lattice Schwinger model using adiabatic
state preparation which, in turn, allows us to compute an expectation value of
the fermion mass operator with respect to the vacuum. Upon taking a continuum
limit we find that our result in massless case agrees with the known exact
result. In massive case, we find an agreement with mass perturbation theory in
small mass regime and deviations in large mass regime. We estimate
computational costs required to take a reasonable continuum limit. Our results
imply that digital quantum simulation is already useful tool to explore
non-perturbative aspects of gauge theories with real time and topological
terms.