Related papers: Grover walks on unitary Cayley graphs and integral…
This paper explains the periodicity of the Grover walk on finite graphs. We characterize the graphs to induce 2, 3, 4, 5-periodic Grover walk and obtain a necessary condition of the graphs to induce an odd-periodic Grover walk.
We consider a quantum walk with two marked vertices, sender and receiver, and analyze its application to perfect state transfer on complete bipartite graphs. First, the situation with both the sender and the receiver vertex in the same part…
Quantum walks on undirected graphs have been studied using symmetric matrices, such as the adjacency or Laplacian matrix, and many results about perfect state transfer are known. We extend some of those results to oriented graphs. We also…
Characterizations graphs of some classes to induce periodic Grover walks have been studied for recent years. In particular, for the strongly regular graphs, it has been known that there are only three kinds of such graphs. Here, we focus on…
We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph $G$ has perfect state transfer if and only if its quotient $G/\pi$, under any equitable partition $\pi$, has perfect state transfer, we…
Perfect state transfer in graphs is a concept arising from quantum physics and quantum computing. Given a graph $G$ with adjacency matrix $A_G$, the transition matrix of $G$ with respect to $A_G$ is defined as $H_{A_{G}}(t) =…
We study perfect state transfer and multiple state transfer in oriented normal Cayley graphs. We construct examples in a variety of groups, ranging from abelian to nonsolvable, and establish some general restrictions and nonexistence…
Given a graph with Hermitian adjacency matrix $H$, perfect state transfer occurs from vertex $a$ to vertex $b$ if the $(b,a)$-entry of the unitary matrix $\exp(-iHt)$ has unit magnitude for some time $t$. This phenomenon is relevant for…
We study perfect state transfer in Kendon's model of discrete quantum walks. In particular, we give a characterization of perfect state transfer purely in terms of the graph spectra, and construct an infinite family of $4$-regular circulant…
Perfect state transfer between two marked vertices of a graph by means of discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit…
A non-complete graph is \emph{$2$-distance-transitive} if, for $i=1,2$ and for any two vertex pairs $(u_1,v_1)$ and $(u_2,v_2)$ with the same distance $i$ in the graph, there exists an element of the graph automorphism group that maps…
Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)},\;t\in\Rl$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and…
Let $X$ be a graph on $n$ vertices with with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer}from $u$ to $v$ occurs if there is a…
Let $L$ denote the Laplacian matrix of a graph $G$. We study continuous quantum walks on $G$ defined by the transition matrix $U(t)=\exp\left(itL\right)$. The initial state is of the pair state form, $e_a-e_b$ with $a,b$ being any two…
In this paper, we characterize perfect state transfer in Cayley graphs for abelian groups that have a cyclic Sylow-2-subgroup. This generalizes a result of Ba\v{s}i\'c from 2013 where he provides a similar characterization for Cayley graphs…
We study the existence of quantum state transfer on non-integral circulant graphs. We find that continuous time quantum walks on quantum networks based on certain circulant graphs with $2^k$ $\left(k\in\mathbb{Z}\right)$ vertices exhibit…
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time…
The question of perfect state transfer existence in quantum spin networks based on weighted graphs has been recently presented by many authors. We give a simple condition for characterizing weighted circulant graphs allowing perfect state…
Quantum walks determined by the coin operator on graphs have been intensively studied. The typical examples of coin operator are the Grover and Fourier matrices. The periodicity of the Grover walk is well investigated. However, the…
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…