Exploring the topological sector optimization on quantum computers

Optimization problems are the core challenge in many fields of science and engineering, yet general and effective methods for finding optimal solutions remain scarce. Quantum computing has been envisioned to help solve such problems, with methods like quantum annealing (QA), grounded in adiabatic evolution, being extensively explored and successfully implemented on quantum simulators such as D-Wave’s annealers and some Rydberg arrays. In this work, we investigate the topological sector optimization (TSO) problem, which has attracted particular interest in the quantum simulation and many-body physics community. We reveal that the topology induced by frustration in the optimization model is an intrinsic obstruction for QA and other traditional methods to approach the ground state. We demonstrate that the difficulties of the TSO problem are not restricted to the gaplessness, but are also due to the topological nature, which was 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 leverages 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 capabilities to address optimization problems are distinct even when considering the quantum computational resources required for practical QITE implementations.