Perfect State Transfer between Non-Antipodal Vertices in Integral Circulant Graphs

Abstract

In this paper, we investigate the existence of perfect state transfer in integral circulant graphs between non-antipodal vertices—vertices that are not at the diameter of a graph. Perfect state transfer is considered on circulant quantum spin networks with nearest-neighbor couplings. The network is described by a circulant graph \(G\), which is characterized by its circulant adjacency matrix \(A\). Formally, we say that there exists perfect state transfer (PST) between vertices \(a, b \in V(G)\) if \(|F(\tau)_{ab}| = 1\) for some positive real number \(\tau\), where \(F(\tau) = \exp(itA)\). Saxena, Severini, and Shparlinski (International Journal of Quantum Information 5 (2007), 417-430) proved that \(|F(\tau)_{aa}| = 1\) for some \(a \in V(G)\) and \(t \in \mathbb{R}\) if and only if all the eigenvalues of \(G\) are integers (that is, the graph is integral). The integral circulant graph \(ICG_n(D)\) has the vertex set \(\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\) and vertices \(a\) and \(b\) are adjacent if \(\gcd(a-b, n) \in D\), where \(D \subseteq \{d: d|n, 1 \leq d \leq n\}\). We characterize completely the class of integral circulant graphs having PST between non-antipodal vertices for \(|D| = 2\). We have thus answered the question posed by Godsil on the existence of classes of graphs with PST between non-antipodal vertices. Moreover, for all values of \(n\) such that \(ICG_n(D)\) has PST (\(n \in 4\mathbb{N}\)), several classes of graphs \(ICG_n(D)\) are constructed such that PST exists between non-antipodal vertices.