2020/03/28 by Sako, Hiroki
#18B99 #46M15 #81P16 #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2003.12732
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 ℤd 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 ∈ ℝ which realizes a given discrete-time quantum walk U. Existence of (V(t))t ∈ ℝ is equivalent to that of a 1-parameter group of automorphisms (V(t))t ∈ ℝ from the object U to U.