  Optimization problems are the core challenge in many fields of science and
engineering, yet general and effective methods are scarce for searching optimal
solutions. Quantum computing has been envisioned to help solve such problems,
for example, the quantum annealing (QA) method based on adiabatic evolution has
been extensively explored and successfully implemented on quantum simulators
such as D-wave's annealers and some Rydberg arrays. In this work, we
investigate topological sector optimization (TSO) problem, which attracts
particular interests in the quantum many-body physics community. We reveal that
the topology induced by frustration in the spin model is an intrinsic
obstruction for QA and other traditional methods to approach the ground state.
We demonstrate that the optimization difficulties of TSO problem are not
restricted to the gaplessness, but are also due to the topological nature which
are often ignored for the analysis of optimization problems before. To solve
TSO problems, we utilize quantum imaginary time evolution (QITE) with a
possible realization on quantum computers, which exploits the property of
quantum superposition to explore the full Hilbert space and can thus address
optimization problems of topological nature. We report the performance of
different quantum optimization algorithms on TSO problems and demonstrate that
their capability to address optimization problems are distinct even when
considering the quantum computational resources required for practical QITE
implementations.
