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：摘要：本文探讨了量子退火中避免能级交叉现象，该框架是量子计算领域有前景的方案，可能为特定任务提供量子优势。量子退火通过使量子系统遵循薛定谔方程演化，最终对末态进行测量以获得优化问题的最优解。然而，量子退火的连续本质使得解析分析具有挑战性，特别是针对瞬时本征能量的分析。绝热定理提供了以高概率获得最优解所需退火时间的理论结果，该时间与最小谱间隙的平方成反比。避免能级交叉可能产生指数级闭合的能隙，进而导致优化问题的运行时间呈指数级增长。本文采用微扰展开推导出退火过程中避免能级交叉发生的条件，并将该条件应用于二分图上的MaxCut问题。研究表明正则二分图不会出现指数级小能隙，表明量子退火在此情形下能高效求解MaxCut问题。另一方面，我们证明顶点度的不规则性可使避免能级交叉发生条件得到满足。本文提供数值实验支持这一理论发展，并讨论指数级闭合能隙的出现与量子退火失效之间的关联。