论文标题
量子步行类别
Categories of quantum walks
论文作者
论文摘要
我们建议类别为$ 1 $维度和多维量子步行。在类别中，一个物体是量子步行，而形态主义是两个量子步行之间的交织操作员。新框架使我们能够以统一的方式讨论量子行走。本文的目的是：（1）我们在新框架中重新解释已知结果。 （2）我们展示了几个新定理。例如，证明每个初始单位向量的每个空间均匀的时间周期分析量子步行都有$ \ mathbb {z}^d $的速度分布的限制分布。分析性是非常弱的状况。 （3）我们询问是否存在连续的量子步行$（v^{（t）}）_ {t \ in \ mathbb {r}} $，它实现了给定的离散时间量子walk $ u $。 $（v^{（t））的存在_ {t \ in \ mathbb {r}} $的存在与$ 1 $ - 参数的自动形态$（v^{（t）}）_ {t \ in \ in \ nathbb {r}} $的$ 1 $ - 参数组。
We propose categories of $1$-dimensional and multi-dimensional quantum walks. In the categories, an object is a quantum walk, and a morphism is an intertwining operator between two quantum walks. The new framework enables us to discuss quantum walks in a unified way. The purposes of this paper are the following: (1) We reinterpret known results in our new framework. (2) We show several new theorems. For example, it is proved that every space-homogeneous time-periodic analytic quantum walk on $\mathbb{Z}^d$ has a limit distribution of velocity for every initial unit vector. Analyticity is a very weak condition. (3) We ask whether there exists a continuous-time quantum walk $(V^{(t)})_{t \in \mathbb{R}}$ which realizes a given discrete-time quantum walk $U$. Existence of $(V^{(t)})_{t \in \mathbb{R}}$ is equivalent to that of a $1$-parameter group of automorphisms $(V^{(t)})_{t \in \mathbb{R}}$ from the object $U$ to $U$.
