懒散的量子步行者是NOT懒惰的
The Lackadaisical Quantum Walker is NOT Lazy at all
摘要：
本文研究了直线上懒洋洋量子游动的性质。该模型首先在~ \cite{wong2015grover}中提出，作为懒惰随机游动的量子模拟，其中每个顶点都连接有$τ$自环。我们推导出了无限步后步行者在原点的定位概率的解析表达式，并得到了步行者的峰值速度。我们还严格地计算了从原点出发的步行者的波函数，得到了整个概率密度函数的长时间近似。作为密度函数的一个应用，我们证明了任意$τ$的懒洋洋量子行走是弹道式传播的，并给出了行走概率分布方差的解析解.
英文摘要：
In this paper, we study the properties of lackadaisical quantum walks on a line. This model is first proposed in~\cite{wong2015grover} as a quantum analogue of lazy random walks where each vertex is attached $τ$ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate rigorously the wave function of the walker starting from the origin and obtain a long time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary $τ$, and give an analytic solution for the variance of the walker's probability distribution.
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