Discrete Quantum Walks on the Symmetric Group

TL;DR

This paper explores discrete quantum walks on symmetric groups using non-commutative Fourier analysis, providing insights into their spectral properties and limiting behavior, which are relevant for quantum algorithm development.

Contribution

It introduces a path integral approach to analyze discrete quantum walks on Cayley graphs of the symmetric group, addressing non-commutativity challenges.

Findings

Characterization of quantum walks on symmetric groups

Analytical description of spectral properties

Insights into limiting behavior of quantum walks

Abstract

The theory of random walks on finite graphs is well developed with numerous applications. In quantum walks, the propagation is governed by quantum mechanical rules; generalizing random walks to the quantum setting. They have been successfully applied in the development of quantum algorithms. In particular, to solve problems that can be mapped to searching or property testing on some specific graph. In this paper we investigate the discrete time coined quantum walk (DTCQW) model using tools from non-commutative Fourier analysis. Specifically, we are interested in characterizing the DTCQW on Cayley graphs generated by the symmetric group ($\sym$) with appropriate generating sets. The lack of commutativity makes it challenging to find an analytical description of the limiting behavior with respect to the spectrum of the walk-operator. We determine certain characteristics of these walks using a path integral approach over the characters of $\sym$.