Characterization of quantum circulant networks having perfect state transfer

Abstract

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 ${a,b\in V(G)}$ if |F(τ) _ab | = 1, for some positive real number τ, where F(t) = exp(i At). Saxena et al. (Int J Quantum Inf 5:417–430, 2007) proved that |F(τ) _aa | = 1 for some ${a\in V(G)}$ and ${\tau\in \mathbb {R}^+}$ 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 ${n\in 4\mathbb {N}}$ and ${D=\widetilde{D_3} \cup D_2\cup 2D_2\cup 4D_2\cup \{n/2^a\}}$ , where ${\widetilde{D_3}=\{d\in D\ |\ n/d\in 8\mathbb {N}\}, D_2= \{d\in D\ |\ n/d\in 8\mathbb {N}+4\}{\setminus}\{n/4\}}$ and ${a\in\{1,2\}}$ . We have thus answered the question of complete characterization of perfect state transfer in integral circulant graphs raised in Angeles-Canul et al. (Quantum Inf Comput 10(3&4):0325–0342, 2010). 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.