Strong dispersion property for the quantum walk on the hypercube

TL;DR

This paper proves that a quantum walk on an n-dimensional hypercube disperses rapidly, making the probability of being at any specific vertex after O(n) steps exponentially small, which is a stronger result than previous mixing analyses.

Contribution

It establishes a strong dispersion property for the quantum walk on the hypercube, providing a rigorous proof that the probability at any vertex is exponentially small after O(n) steps.

Findings

Probability at any vertex after O(n) steps is O(1.4818^{-n})

Improves understanding of quantum walk mixing properties

Uses analytic properties of Bessel functions in the proof

Abstract

We show that the discrete time quantum walk on the Boolean hypercube of dimension $n$ has a strong dispersion property: if the walk is started in one vertex, then the probability of the walker being at any particular vertex after $O(n)$ steps is of an order $O(1.4818^{-n})$. This improves over the known mixing results for this quantum walk which show that the probability distribution after $O(n)$ steps is close to uniform but do not show that the probability is small for every vertex. A rigorous proof of this result involves an intricate argument about analytic properties of Bessel functions.