In this work we focus on the notion of quantum hitting time for discrete-time Szegedy quantum walks, compared to its classical counterpart. Under suitable hypotheses, quantum hitting time is known to be of the order of the square root of classical hitting time: this quadratic speedup is a remarkable example of the computational advantages associated with quantum approaches. Our purpose here is twofold. On one hand, we provide a detailed proof of quadratic speedup for time-reversible walks within the Szegedy framework, in a language that should be familiar to the linear algebra community. Moreover, we explore the use of a general distribution in place of the stationary distribution in the definition of quantum hitting time, through theoretical considerations and numerical experiments.

翻译：本文聚焦于离散时间Szegedy量子游走的量子击中时间概念，并与经典对应概念进行比较。在适当假设下，已知量子击中时间量级为经典击中时间的平方根：这种二次加速是量子方法带来计算优势的显著例证。本文旨在两个方面展开：一方面，我们为Szegedy框架下时间可逆游走的二次加速提供详细证明，采用线性代数领域熟悉的语言进行阐述；另一方面，我们通过理论分析与数值实验，探讨在量子击中时间定义中使用一般分布替代平稳分布的可能性。