  Quantum complexity is conjectured to probe inside of black hole horizons (or
wormhole) via gauge gravity correspondence. In order to have a better
understanding of this correspondence, we study time evolutions of complexities
for generic Abelian pure gauge theories. For this purpose, we discretize $U(1)$
gauge group as $\mathbf{Z}_N$ and also continuum spacetime as lattice
spacetime, and this enables us to define a universal gate set for these gauge
theories, and evaluate time evolutions of the complexities explicitly. We find
that for a generic class of diagonal Hamiltonians to achieve a large complexity
$\sim \exp(\mbox{entropy})$, which is one of the conjectured criteria necessary
to have a dual black hole, the Abelian gauge theory needs to be maximally
nonlocal.
