Perfect state transfer in cubelike graphs

Suppose C is a subset of non-zero vectors from the vector space . The cubelike graph X(C) has as its vertex set, and two elements of are adjacent if their difference is in C. If M is the d×|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 π/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 τ=π/2D, where D is the greatest common divisor of the weights of the code words. We show that perfect state transfer occurs at time π/4 if and only if D=2 and the code is self-orthogonal.