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Quantum Information and Computation

Stationary measures of QW on cycles

Takeshi Kajiwara, Norio Konno

Published Nov 1, 2024• 0 citations
Publisher

Abstract

We introduce eigen-independency of certain unitary matrices of degree 2. We show that this property characterizes stationary measures of quantum walks on cycles with odd number of vertices. We also characterize the stationary measures of the two-state Hadamard walk (both moving-shift and flip-flop shit) on cycles with $p$ vertices, where $p$ is an odd prime number.

DOI
10.26421/qic24.13-14-4

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We introduce eigen-independency of certain unitary matrices of degree 2. We show that this property characterizes stationary measures of quantum walks on cycles with odd number of vertices. We also characterize the stationary measures of the two-state Hadamard walk (both moving-shift and flip-flop shit) on cycles with ppp vertices, where ppp is an odd prime number.
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We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 ×\times× 2 unitary matrix UUU. In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when UUU is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.
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.
The present paper treats the period TN of the Hadamard walk on a cycle CN with N vertices. Dukes (2014) considered the periodicity of more general quantum walks on CN and showed T2=2, T4=8, T8=24 for the Hadamard walk case. We prove that the Hadamard walk does not have any period except for his case, i.e., N = 2,4,8. Our method is based on a path counting and cyclotomic polynomials which is different from his approach based on the property of eigenvalues for unitary matrix that determines the evolution of the walk.
We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 times 2 unitary matrix U. In our previous paper [15], we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except U is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.
Quantum walks, the quantum counterpart of classical random walks, are extensively studied for their applications in mathematics, quantum physics, and quantum information science. This study explores the periods and absolute zeta functions of Grover walks on cycle graphs. Specifically, we investigate Grover walks with an odd number of states and determine their periods for cycles with any number of vertices greater than or equal to two. In addition, we compute the absolute zeta functions of M-type Grover walks with finite periods. These results advance the understanding of the properties of Grover walks and their connection to absolute zeta functions.