Exact solutions and symmetry analysis for the limiting probability distribution of quantum walks

TL;DR

This paper derives exact solutions for the limiting probability distribution of one-dimensional quantum walks on cycles with general coins and shift operators, analyzing symmetry conditions and dynamic properties.

Contribution

It provides the first exact solutions for these quantum walks with general operators and explores symmetry generation and related dynamic measures.

Findings

Exact solutions for limiting distributions are derived.

Conditions for symmetric quantum walks are identified.

Deviation and mixing times are analyzed.

Abstract

In the literature, there are numerous studies of one-dimensional discrete-time quantum walks (DTQWs) using a moving shift operator. However, there is no exact solution for the limiting probability distributions of DTQWs on cycles using a general coin or swapping shift operator. In this paper, we derive exact solutions for the limiting probability distribution of quantum walks using a general coin and swapping shift operator on cycles for the first time. Based on the exact solutions, we show how to generate symmetric quantum walks and determine the condition under which a symmetric quantum walk appears. Our results suggest that choosing various coin and initial state parameters can achieve a symmetric quantum walk. By defining a quantity to measure the variation of symmetry, deviation and mixing time of symmetric quantum walks are also investigated.