  Simulating lattice gauge theory (LGT) Hamiltonian and its nontrivial states
by programmable quantum devices has attracted numerous attention in recent
years. Rydberg atom arrays constitute one of the most rapidly developing arenas
for quantum simulation and quantum computing. The $\mathbb{Z}_2$ LGT and
topological order has been realized in experiments while the $U(1)$ LGT is
being worked hard on the way. States of LGT have local constraint and are
fragmented into several winding sectors with topological protection. It is
therefore difficult to reach the ground state in target sector for experiments,
and it is also an important task for quantum topological memory. Here, we
propose a protocol of sweeping quantum annealing (SQA) for searching the ground
state among topological sectors. With the quantum Monte Carlo method, we show
that this SQA has linear time complexity of size with applications to the
antiferromagnetic transverse field Ising model, which has emergent $U(1)$ gauge
fields. This SQA protocol can be realized easily on quantum simulation
platforms such as Rydberg array and D-wave annealer. We expect this approach
would provide an efficient recipe for resolving the topological hindrances in
quantum optimization and the preparation of quantum topological state.
