Perfect state transfer using Markovian quantum walk

TL;DR

This paper introduces a novel method using Markovian quantum walks to achieve perfect state transfer (PST) in various graph structures, surpassing limitations of continuous-time quantum walks.

Contribution

It demonstrates that Markovian quantum walks enable PST in path and cycle graphs of arbitrary size, expanding the scope of quantum communication protocols.

Findings

PST established between extreme vertices of any path graph.

Any symmetric pair of vertices in a path graph allows PST.

Cycle graphs with more than 4 vertices do not allow PST via continuous-time quantum walk, but do with Markovian quantum walk.

Abstract

The quantum Perfect State Transfer (PST) is a fundamental tool of quantum communication in a network. It is not easy to achieve in practice. The original idea of PST depends on the fundamentals of the continuous-time quantum walk. A path graph with at most three vertices allows PST based on continuous-time quantum walk. Based on the Markovian quantum walk, we introduce a significantly powerful method for PST in this article. We establish PST between the extreme vertices of a path graph of arbitrary length. Moreover, any pair of symmetric vertices in a path graph allows PST under Markovian quantum walks. We extend our investigations for the cycle graphs. The cycle graphs with more than $4$ vertices do not allow the PST based on the continuous-time quantum walk. In contrast, a cycle graph with $2m$ vertices exhibits PST based on Markovian quantum walk between the vertices $j$ and $j + m$ for $j = 0, 1, \dots (m - 1)$, where $m > 0$ is an integer.