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- Title
Characterization of quantum circulant networks having perfect state transfer.
- Authors
Bašić, Milan
- 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)| = 1, for some positive real number τ, where F( t) = exp(i At). Saxena et al. (Int J Quantum Inf 5:417-430, ) proved that | F( τ)| = 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 ( D) has the vertex set Z = {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 ( 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, ). 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.
- Subjects
QUANTUM theory; NUCLEAR spin; EXISTENCE theorems; EIGENVALUES; QUANTUM communication; GRAPH theory; SYMMETRY (Physics)
- Publication
Quantum Information Processing, 2013, Vol 12, Issue 1, p345
- ISSN
1570-0755
- Publication type
Article
- DOI
10.1007/s11128-012-0381-z