  We provide exact results for the dynamics of local-operator entanglement in
quantum circuits with two-dimensional wires featuring ultralocal solitons, i.e.
single-site operators which, up to a phase, are simply shifted by the time
evolution. We classify all circuits allowing for ultralocal solitons and show
that only dual-unitary circuits can feature moving ultralocal solitons. Then,
we rigorously prove that if a circuit has an ultralocal soliton moving to the
left (right), the entanglement of local operators initially supported on even
(odd) sites saturates to a constant value and its dynamics can be computed
exactly. Importantly, this does not bound the growth of complexity in chiral
circuits, where solitons move only in one direction, say to the left. Indeed,
in this case we observe numerically that operators on the odd sublattice have
unbounded entanglement. Finally, we present a closed-form expression for the
local-operator entanglement entropies in circuits with ultralocal solitons
moving in both directions. Our results hold irrespectively of integrability.
