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 perfect state transfer (PST) between vertices a, b ∈ V (G) if |F (τ)ab| = 1, for some positive real number τ , where F (t) = exp(ıAt). Saxena, Severini and Shparlinski (International Journal of Quantum Information 5 (2007), 417–430) proved that |F (τ)aa| = 1 for some a ∈ V (G) and τ ∈ R if and only if all eigenvalues of G are integer (that is, the graph is integral). The integral circulant graph ICGn(D) has the vertex set Zn = {0, 1, 2, . . . , n − 1} and vertices a and b are adjacent if gcd(a − b, n) ∈ D, where D ⊆ {d : d | n, 1 ≤ d < n}. These graphs are highly symmetric and have important applications in chemical graph theory. We show that ICGn(D) has PST if and only if n ∈ 4N and D = D̃3 ∪D2 ∪ 2D2 ∪ 4D2 ∪ {n/2a}, where D̃3 = {d ∈ D | n/d ∈ 8N}, D2 = {d ∈ D | n/d ∈ 8N + 4} \ {n/4} and a ∈ {1, 2}. We have thus answered the question of complete characterization of perfect state transfer in integral circulant graphs raised in Quantum Information and Computation, Vol. 10, No. 3&4 (2010) 0325-0342 by Angeles-Canul 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. AMS Subj. Class.: 05C12, 05C50