Characterization of circulant graphs having perfect state transfer

In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph $G$, which is characterized by its circulant adjacency matrix $A$. Formally, we say that there exists a {\it perfect state transfer} (PST) between vertices $a,b\in V(G)$ if $\mathrm{\mid }F(\tau {)}_{ab}\mathrm{\mid }=1$, for some positive real number $\tau$, where $F(t)=\exp (\text{i}At)$. Saxena, Severini and Shparlinski ({\it International Journal of Quantum Information} 5 (2007), 417–430) proved that $\mathrm{\mid }F(\tau {)}_{aa}\mathrm{\mid }=1$ for some $a\in V(G)$ and $\tau\in \R^+$ifandonlyifalleigenvaluesof$G$ are integer (that is, the graph is integral). The integral circulant graph ${\text{ICG}}_n(D)$ has the vertex set $Z_n = {0, 1, 2, …, n - 1}$andvertices$a$and$b$areadjacentif$\gcd(a-b,n)\in D$,where$D \subseteq {d : d \mid n,\ 1\leq d<n}$. These graphs are highly symmetric and have important applications in chemical graph theory. We show that ${\text{ICG}}_n(D)$ has PST if and only if $n\in 4\mathbb{N}$ and $D=\widetilde{D_3}\cup D_2\cup 2D_2\cup 4D_2\cup \{n\mathrm{/}{2}^a\}$, where $\widetilde{D_3}=\{d\in D\mathrm{\mid }n\mathrm{/}d\in 8\mathbb{N}\}$, $D_2= {d\in D\ |\ n/d\in 8\N+4}\setminus {n/4}$and$a\in{1,2}$. We have thus answered the question of complete characterization of perfect state transfer in integral circulant graphs raised in {\it Quantum Information and Computation}, Vol. 10, No. 3&4 (2010) 0325–0342 by Angeles-Canul {\it et al.} Furthermore, we also calculate perfect quantum communication distance (distance between vertices where PST occurs) and describe the spectra of integral circulant graphs having PST. We conclude by giving a closed form expression calculating the number of integral circulant graphs of a given order having PST.