Categories of quantum walks
Abstract
We propose categories of $1$dimensional and multidimensional 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 spacehomogeneous timeperiodic 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 continuoustime quantum walk $(V^{(t)})_{t \in \mathbb{R}}$ which realizes a given discretetime 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 $U$.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.12732
 arXiv:
 arXiv:2003.12732
 Bibcode:
 2020arXiv200312732S
 Keywords:

 Mathematical Physics;
 Mathematics  Category Theory;
 18B99;
 46M15;
 81P16