A new type of quantum walks based on decomposing quantum states

TL;DR

This paper introduces a novel 2-state decomposed quantum walk model that can replicate 4-state quantum walks, revealing new localization and spreading behaviors and establishing a connection to the Grover walk.

Contribution

The paper presents a new 2-state decomposed quantum walk model that can simulate 4-state quantum walks, including the Grover walk, with derived limit theorems.

Findings

DQWs can exhibit both localization and spreading phenomena.

A 2-state DQW can be equivalent to certain 4-state QWs.

Derived weak limit theorem for DQWs related to 4-state QWs.

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.