Perfect state transfer in cubelike graphs

✍ Scribed by Wang-Chi Cheung; Chris Godsil

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 177 KB
• Volume: 435
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2011.04.022

✦ Synopsis

Suppose C is a subset of non-zero vectors from the vector space Z d 2 .

The cubelike graph X(C) has Z d 2 as its vertex set, and two elements

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.