Simple Quantum Coins Enable Pretty Good State Transfer on Every Hypercube

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.