Uniform Stationary Measure of Space-Inhomogeneous One-Dimensional Three-State Quantum Walks

Yusuke, Ide, 40105, Norio Konno +3 more

Caishi Wang, Xiangying Lu, Wen-Ling Wang

Chusei Kiumi, Kei Saito

Ide, Yusuke, Norio Konno, Nakayama Daichi

The discrete-time quantum walk (QW) is a quantum version of the random walk (RW) and has been widely investigated for the last two decades. Some remarkable properties of QW are well known. For example, QW has a ballistic spreading, i.e., QW is quadratically faster than RW. For some cases, localization occurs: a walker stays at the starting position forever. In this paper, we consider stationary measures of two-state QWs on the line. It was shown that for any space-homogeneous model, the uniform measure becomes the stationary measure. However, the corresponding result for space-inhomogeneous model is not known. Here, we present a class of space-inhomogeneous QWs on the line and cycles in which the uniform measure is stationary. Furthermore, we briefly discuss a difference between QWs and RWs.

Shengsheng Liu, Caishi Wang, Jijun Zhao

Of a quantum walk, its stationary measures play an important role in understanding its evolution behavior. In this paper we investigate stationary measures of two models of space-inhomogeneous three-state quantum walk on the line. By using the method of reduced matrix, we find out stationary measures of the two models under some mild conditions. Our results generalize the corresponding ones existing in the literature.

Hikari Kawai, Komatsu, Takashi, Norio Konno

In this paper, we consider stationary measures of discrete-time three-state quantum walks including the Fourier and Grover walks in the one-dimensional lattice. We give non-uniform stationary measures by solving the corresponding eigenvalue problem. Our new method is based on a reduced matrix, which is different from the generating function approach in our previous work. As a corollary, the Fourier walk on the cycle has a stationary measure with a periodicity.