3,210 research outputs found
Quantum Walks on Strongly Regular Graphs
This thesis studies the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs.
We begin by finding the eigenvalues of matrices describing the quantum walk for regular graphs. We also show that if two graphs are isomorphic, then the corresponding matrices produced by the procedure of Emms et al. are cospectral. We then look at the entries of the cube of the transition matrix and find an expression for the matrices produced by the procedure of Emms et al. in terms of the adjacency matrix and incidence matrices of the graph
Discrete Quantum Walks on Graphs and Digraphs
This thesis studies various models of discrete quantum walks on graphs and digraphs via a spectral approach.
A discrete quantum walk on a digraph is determined by a unitary matrix , which acts on complex functions of the arcs of . Generally speaking, is a product of two sparse unitary matrices, based on two direct-sum decompositions of the state space. Our goal is to relate properties of the walk to properties of , given some of these decompositions.
We start by exploring two models that involve coin operators, one due to Kendon, and the other due to Aharonov, Ambainis, Kempe, and Vazirani. While is not defined as a function in the adjacency matrix of the graph , we find exact spectral correspondence between and . This leads to characterization of rare phenomena, such as perfect state transfer and uniform average vertex mixing, in terms of the eigenvalues and eigenvectors of . We also construct infinite families of graphs and digraphs that admit the aforementioned phenomena.
The second part of this thesis analyzes abstract quantum walks, with no extra assumption on . We show that knowing the spectral decomposition of leads to better understanding of the time-averaged limit of the probability distribution. In particular, we derive three upper bounds on the mixing time, and characterize different forms of uniform limiting distribution, using the spectral information of .
Finally, we construct a new model of discrete quantum walks from orientable embeddings of graphs. We show that the behavior of this walk largely depends on the vertex-face incidence structure. Circular embeddings of regular graphs for which has few eigenvalues are characterized. For instance, if has exactly three eigenvalues, then the vertex-face incidence structure is a symmetric -design, and is the exponential of a scalar multiple of the skew-symmetric adjacency matrix of an oriented graph. We prove that, for every regular embedding of a complete graph, is the transition matrix of a continuous quantum walk on an oriented graph
Perfect state transfer, graph products and equitable partitions
We describe new constructions of graphs which exhibit perfect state transfer
on continuous-time quantum walks. Our constructions are based on variants of
the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products
(which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If
is a graph with perfect state transfer at time , where t_{G}\Spec(G)
\subseteq \ZZ\pi, and is a circulant with odd eigenvalues, their weak
product has perfect state transfer. Also, if is a regular
graph with perfect state transfer at time and is a graph where
t_{H}|V_{H}|\Spec(G) \subseteq 2\ZZ\pi, their lexicographic product
has perfect state transfer. (2) The double cone on any
connected graph , has perfect state transfer if the weights of the cone
edges are proportional to the Perron eigenvector of . This generalizes
results for double cone on regular graphs studied in
[BCMS09,ANOPRT10,ANOPRT09]. (3) For an infinite family \GG of regular graphs,
there is a circulant connection so the graph K_{1}+\GG\circ\GG+K_{1} has
perfect state transfer. In contrast, no perfect state transfer exists if a
complete bipartite connection is used (even in the presence of weights)
[ANOPRT09]. We also describe a generalization of the path collapsing argument
[CCDFGS03,CDDEKL05], which reduces questions about perfect state transfer to
simpler (weighted) multigraphs, for graphs with equitable distance partitions.Comment: 18 pages, 6 figure
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