Categories of quantum walks
Abstract
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 $U$.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2020
- DOI:
- 10.48550/arXiv.2003.12732
- arXiv:
- arXiv:2003.12732
- Bibcode:
- 2020arXiv200312732S
- Keywords:
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- Mathematical Physics;
- Mathematics - Category Theory;
- 18B99;
- 46M15;
- 81P16