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 $|F(\tau)_{ab}|=1$, for some positive real number $\tau$, where $F(t)=\exp(\i At)$. Saxena, Severini and Shparlinski ({\it International Journal of Quantum Information} 5 (2007), 417--430) proved that $|F(\tau)_{aa}|=1$ for some $a\in V(G)$ and $\tau\in \R^+$ if and only if all eigenvalues of $G$ are integer (that is, the graph is integral). The integral circulant graph $\ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, ..., n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, 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 $\ICG_n (D)$ has PST if and only if $n\in 4\N$ and $D=\widetilde{D_3}\cup D_2\cup 2D_2\cup 4D_2\cup \{n/2^a\}$, where $\widetilde{D_3}=\{d\in D\ |\ n/d\in 8\N\}$, $D_2= \{d\in D\ |\ n/d\in 8\N+4\}\setminus \{n/4\}$ 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.