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 \documentclass[12pt]{minimal}
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\begin{document}$${a,b\in V(G)}$$\end{document} 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 \documentclass[12pt]{minimal}
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\begin{document}$${a\in V(G)}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${\tau\in \mathbb {R}^+}$$\end{document} 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 \documentclass[12pt]{minimal}
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\begin{document}$${\gcd(a-b,n)\in D}$$\end{document} , where \documentclass[12pt]{minimal}
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\begin{document}$${D \subseteq \{d : d \mid n, \ 1 \leq d < n\}}$$\end{document} . These graphs are highly symmetric and have important applications in chemical graph theory. We show that ICGn (D) has PST if and only if \documentclass[12pt]{minimal}
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\begin{document}$${n\in 4\mathbb {N}}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${D=\widetilde{D_3} \cup D_2\cup 2D_2\cup 4D_2\cup \{n/2^a\}}$$\end{document} , where \documentclass[12pt]{minimal}
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\begin{document}$${\widetilde{D_3}=\{d\in D\ |\ n/d\in 8\mathbb {N}\}, D_2= \{d\in D\ |\ n/d\in 8\mathbb {N}+4\}{\setminus}\{n/4\}}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${a\in\{1,2\}}$$\end{document} . 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.
