This paper explores the phenomenon of avoided level crossings in quantum annealing, a promising framework for quantum computing that may provide a quantum advantage for certain tasks. Quantum annealing involves letting a quantum system evolve according to the Schr\"odinger equation, with the goal of obtaining the optimal solution to an optimization problem through measurements of the final state. However, the continuous nature of quantum annealing makes analytical analysis challenging, particularly with regard to the instantaneous eigenenergies. The adiabatic theorem provides a theoretical result for the annealing 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 lead to exponentially long running times for optimization problems. In this paper, we use a perturbative expansion to derive a condition for the occurrence of an avoided level crossing during the annealing process. We then apply this condition to the MaxCut problem on bipartite graphs. We show that no exponentially small gaps arise for regular bipartite graphs, implying that QA can efficiently solve MaxCut in that case. On the other hand, we show that irregularities in the vertex degrees can lead to the satisfaction of the avoided level crossing occurrence condition. We provide numerical evidence to support this theoretical development, and discuss the relation between the presence of exponentially closing gaps and the failure of quantum annealing.

翻译：本文研究了量子退火中能级避让交叉现象。量子退火是一种具有前景的量子计算框架，有望在特定任务中展现量子优势。该过程使量子系统遵循薛定谔方程演化，旨在通过测量最终态获得优化问题的最优解。然而，量子退火的连续性特征使得解析分析极具挑战，尤其体现在瞬时本征能量的分析上。绝热定理给出了以高概率获得最优解所需的退火时间理论结果，该时间与最小谱能隙的平方成反比。避让交叉会导致能隙呈指数级闭合，进而使得优化问题的求解时间呈指数级增长。本文采用微扰展开方法推导了退火过程中避让交叉现象发生的条件，并将该条件应用于二部图上的MaxCut问题。研究表明，正则二部图中不会出现指数级小能隙，这意味着量子退火可高效求解该类图中的MaxCut问题。另一方面，我们证明顶点度的非正则性可能满足避让交叉的发生条件。本文提供数值实验佐证该理论发现，并论述指数级闭合能隙与量子退火失效之间的关联。