Quantum walk in symmetric Cayley graph over $\Z_2^n$

TL;DR

This paper demonstrates that discrete quantum walks on symmetric Cayley graphs over
72_2^n have polynomial hitting and returning times with high probability, supported by new Kravchuk coefficient estimates, and proposes a probabilistic algorithm for finding antipodal vertices.

Contribution

It introduces a new analysis of quantum walk hitting times on symmetric Cayley graphs and provides a probabilistic algorithm for antipodal vertex search.

Findings

Quantum walk hitting time is polynomial in graph degree.

Return time of quantum walk is polynomial with high probability.

A probabilistic polynomial algorithm for antipodal vertex search is presented.

Abstract

We show that the hitting time of the discrete quantum walk on a symmetric Cayley graph over $\Z_2^n $ from a vertex to its antipodal is polynomial in degree of the graph. We prove that returning time of quantum walk on a symmetric Cayley graph over $\Z_2^n $ is polynomial and the probability to hit is almost one. To prove it, we give a new estimation of Kravchuk coefficients. We give an example of a probabilistic polynomial algorithm that finds an antipodal vertex in symmetric Cayley graphs.