Perfect state transfer on abelian Cayley graphs

Tan, Yingying; Feng, Keqin; Cao, Xiwang

doi:10.1016/j.laa.2018.11.011

Cited by 37 publications

(32 citation statements)

References 29 publications

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“…The phenomenon of perfect state transfer (PST) in quantum communication networks was originally introduced by Bose in [14]. This work has attracted much research interest since many applications have been found in quantum information processing and cryptography (see [1,2,3,5,11,15,17,18,34,38,37] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Perfect State Transfer on Cayley Graphs over Dihedral Groups: The Non-Normal Case

Cao¹,

Chen

Ling³

2020

Electron. J. Combin.

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Recently, perfect state transfer (PST for short) on graphs has attracted great attention due to their applications in quantum information processing and quantum computations. Many constructions and results have been established through various graphs. However, most of the graphs previously investigated are abelian Cayley graphs. Necessary and sufficient conditions for Cayley graphs over dihedral groups having perfect state transfer were studied recently. The key idea in that paper is the assumption of the normality of the connection set. In those cases, viewed as an element in a group algebra, the connection set is in the center of the group algebra, which makes the situations just like in the abelian case. In this paper, we study the non-normal case. In this case, the discussion becomes more complicated. Using the representations of the dihedral group $D_n$, we show that ${\rm Cay}(D_n,S)$ cannot have PST if $n$ is odd. For even integers $n$, it is proved that if ${\rm Cay}(D_n,S)$ has PST, then $S$ is normal.

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Section: Introductionmentioning

confidence: 99%

Perfect State Transfer on Cayley Graphs over Dihedral Groups: The Non-Normal Case

Cao¹,

Chen

Ling³

2020

Electron. J. Combin.

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“…We also remark that Cayley graphs of the group Z n 2 , known as cubelike graphs, have received much attention previously [6,15]. As the quotient of an extraspecial group of order 2 2n+1 by its center of order 2 is isomorphic to Z 2n 2 , it is interesting to compare the quantum walks on these groups.…”

Section: Discussionmentioning

confidence: 98%

“…Bernasconi et al and Cheung et al studied perfect state transfer in cubelike graphs [3,6]. Tan et al generalized this work by studying Cayley graphs of arbitrary abelian groups in [15], and provided conditions for perfect state transfer in terms of the spectrum.…”

Section: Introductionmentioning

confidence: 99%

Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups

Sin¹,

Sorci²

2020

Preprint

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We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian p-groups, called extraspecial 2-groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state transfer and fractional revival, and use partial spreads to construct graphs admitting these various phenomena. Lastly, we use a result of Ada Chan to show that there is no normal Cayley graph of an extraspecial p-group that admits instantaneous uniform mixing.

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“…We extend Theorem 2.4 of [22] to (Laplacian) fractional revival here. (1) The eigenvalues of L(G) are integers;…”

Section: Continuous-time Quantum Walksmentioning

confidence: 97%

Complex Hadamard Diagonalisable Graphs

Chan

Fallat

Kirkland

et al. 2020

Preprint

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In light of recent interest in Hadamard diagonalisable graphs (graphs whose Laplacian matrix is diagonalisable by a Hadamard matrix), we generalise this notion from real to complex Hadamard matrices. We give some basic properties and methods of constructing such graphs. We show that a large class of complex Hadamard diagonalisable graphs have vertex sets forming an equitable partition, and that the Laplacian eigenvalues must be even integers. We provide a number of examples and constructions of complex Hadamard diagonalisable graphs, including two special classes of graphs: the Cayley graphs over Z d r , and the non-complete extended p-sum (NEPS). We discuss necessary and sufficient conditions for (α, β)-Laplacian fractional revival and perfect state transfer on continuous-time quantum walks described by complex Hadamard diagonalisable graphs and provide examples of such quantum state transfer.

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Perfect state transfer on abelian Cayley graphs

Cited by 37 publications

References 29 publications

Perfect State Transfer on Cayley Graphs over Dihedral Groups: The Non-Normal Case

Perfect State Transfer on Cayley Graphs over Dihedral Groups: The Non-Normal Case

Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups

Complex Hadamard Diagonalisable Graphs

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