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 ∈ 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 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) ∈ 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 ICG n (D) has PST if and only if n ∈ 4N andand 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.

Quantum state transfer on a class of circulant graphs

“…Apart from cycles, we have also found families of non-integral circulant graphs exhibiting PGST in [15,18]. Several other ralavent results regarding state transfer can be found in [2,8,12,17,16,14,19,20]. We extend some pre-existing results appearing in [15,18] on circulant graphs admitting PGST and re-discover families of circulant graphs on 2 k vertices exhibiting PST.…”

“…It is shown in [20] that if a cycle admits PST then it must be integral and this leads to the conclusion that only C 4 exhibits PST (see [2]). We further investigated and in [18], we have found a characterization of all cycles admitting PGST.…”

“…If there is a sub-sequential limit t 0 of {t k } then Equation 2 gives H(t 0 )e u = γe v , in which case we say that there is perfect state transfer (PST) (see [4]) between u and v at time t 0 . If H(t 0 )e u = γe u for t 0 = 0 then G is said to be periodic at u at time t 0 .…”

“…Therefore, by Proposition 1.4, if Cay (Z n , S) exhibits PGST then Cay (Z n , S) also exhibits PST. A complete characterization of circulant graphs exhibiting perfect state transfer is appearing in [2]. Notice that Theorem 2.8 in fact reveals all integral circulant graphs on 2 k vertices exhibiting PST.…”

Exact Inverse Matrices of Fermat and Mersenne Circulant Matrix

“…Distributed differential space-time codes that work for networks with any number of relays using circulant matrices were proposed by Jing and Jafarkhani in [3]. Bašić [4] solved the question for when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. Wang et al considered two-way transmission model ensured that circular convolution between two frequency selective channels in [5].…”