A new type of quantum walks based on decomposing quantum states

Abstract

In this paper, the 2-state decomposed-type quantum walk (DQW) on a line is introduced as an extension of the 2-state quantum walk (QW). The time evolution of the DQW is defined with two different matrices, one is assigned to a real component, and the other is assigned to an imaginary component of the quantum state. Unlike the ordinary 2-state QWs, localization and the spreading phenomenon can coincide in DQWs. Additionally, a DQW can always be converted to the corresponding 4-state QW with identical probability measures. In other words, a class of 4-state QWs can be realized by DQWs with 2 states. In this work, we reveal that there is a 2-state DQW corresponding to the 4-state Grover walk. Then, we derive the weak limit theorem of the class of DQWs corresponding to 4-state QWs which can be regarded as the generalized Grover walks.

Localization of space-inhomogeneous three-state quantum walks

“…Three-state QWs have an interesting property called localization, where the probability of finding the particle around the initial position remains positive in the long-time limit. Localization on multi-state (three or more states) QWs is actively researched by previous studies [10][11][12][13][14][15][16]. In the research of multi-state QWs, the Grover walk, whose time evolution is defined by the Grover matrix as a coin matrix, often plays an essential role.…”