  Recently, novel numerical computation on quantum mechanics by using a
bootstrap method was proposed by Han, Hartnoll, and Kruthoff. We consider
whether this method works in systems with a $\theta$-term, where the standard
Monte-Carlo computation may fail due to the sign problem. As a starting point,
we study quantum mechanics of a charged particle on a circle in which a
constant gauge potential is a counterpart of a $\theta$-term. We find that it
is hard to determine physical quantities as functions of $\theta$ such as
$E(\theta)$, except at $\theta=0$ and $\pi$. On the other hand, the
correlations among observables for energy eigenstates are correctly reproduced
for any $\theta$. Our results suggest that the bootstrap method may work not
perfectly but sufficiently well, even if a $\theta$-term exists in the system.
