Simple Quantum Coins Enable Pretty Good State Transfer on Every Hypercube

TL;DR

This paper introduces weighted Grover coins for coined quantum walks that enable pretty good state transfer on all hypercubes, extending previous results limited to prime dimensions, and provides a general condition for other graphs.

Contribution

It develops weighted Grover coins that achieve pretty good state transfer on all hypercubes, regardless of dimension, and generalizes the approach to other graph structures.

Findings

Weighted Grover coins enable state transfer on all hypercubes.

Modification of only one arc per vertex suffices for transfer.

A sufficient condition for state transfer on other graphs is established.

Abstract

We consider pretty good state transfer in coined quantum walks between antipodal vertices on the hypercube $Q_d$. When $d$ is a prime, this was proven to occur in the arc-reversal walk with Grover coins. We extend this result by constructing weighted Grover coins that enable pretty good state transfer on every $Q_d$. Our coins are real, and require modification of the weight on only one arc per vertex. We also generalize our approach and establish a sufficient condition for pretty good state transfer to occur on other graphs.