Perfect state transfer in cubelike graphs

Abstract

Suppose $C$ is a subset of non-zero vectors from the vector space $\mathbb{Z}_2^d$. The cubelike graph $X(C)$ has $\mathbb{Z}_2^d$ as its vertex set, and two elements of $\mathbb{Z}_2^d$ are adjacent if their difference is in $C$. If $M$ is the $d\times |C|$ matrix with the elements of $C$ as its columns, we call the row space of $M$ the code of $X$. We use this code to study perfect state transfer on cubelike graphs. Bernasconi et al have shown that perfect state transfer occurs on $X(C)$ at time $\pi/2$ if and only if the sum of the elements of $C$ is not zero. Here we consider what happens when this sum is zero. We prove that if perfect state transfer occurs on a cubelike graph, then it must take place at time $\tau=\pi/2D$, where $D$ is the greatest common divisor of the weights of the code words. We show that perfect state transfer occurs at time $\pi/4$ if and only if D=2 and the code is self-orthogonal.