Characterization of quantum circulant networks having perfect state transfer

Characterization of quantum circulant networks having perfect state transfer 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Characterization of quantum circulant networks having perfect state transfer

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Publisher
Springer US
Copyright
Copyright © 2012 by Springer Science+Business Media, LLC
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-012-0381-z
Publisher site
See Article on Publisher Site

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.

Journal

Quantum Information ProcessingSpringer Journals

Published: Mar 4, 2012

References

  • On quantum perfect state transfer in weighted join graphs
    Angeles-Canul, R.J.; Norton, R.M.; Opperman, M.C.; Paribello, C.C.; Russell, M.C.; Tamon, C.
  • Integral circulant graphs
    So, W.

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