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This paper explores the phenomenon of avoided level crossings in quantumannealing, a promising framework for quantum computing that may provide aquantum advantage for certain tasks. Quantum annealing involves letting aquantum system evolve according to the Schr\"odinger equation, with the goal ofobtaining the optimal solution to an optimization problem through measurementsof the final state. However, the continuous nature of quantum annealing makesanalytical analysis challenging, particularly with regard to the instantaneouseigenenergies. The adiabatic theorem provides a theoretical result for theannealing time required to obtain the optimal solution with high probability,which is inversely proportional to the square of the minimum spectral gap.Avoided level crossings can create exponentially closing gaps, which can leadto exponentially long running times for optimization problems. In this paper,we use a perturbative expansion to derive a condition for the occurrence of anavoided level crossing during the annealing process. We then apply thiscondition to the MaxCut problem on bipartite graphs. We show that noexponentially small gaps arise for regular bipartite graphs, implying that QAcan efficiently solve MaxCut in that case. On the other hand, we show thatirregularities in the vertex degrees can lead to the satisfaction of theavoided level crossing occurrence condition. We provide numerical evidence tosupport this theoretical development, and discuss the relation between thepresence of exponentially closing gaps and the failure of quantum annealing.

Anti-crossings occurrence as exponentially closing gaps in Quantum Annealing