Continuous-time quantum walks on graphs

Samenvatting

The quantum analog of a random walk on a graph is appropriately called a quantum walk on a
graph. In these quantum walks, instead of deterministically travelling over the nodes of a graph, we
pass through superpositions of the nodes of the graph. Quantum walks are universal for quantum
algorithms. Moreover, in the same sense their classical counterparts can exponentially speed up
classical computer algorithms, quantum walks have the potential to speed up quantum algorithms
exponentially as well. In this thesis, we will study significant properties of these quantum walks on
graphs like periodicity, perfect state transfer (PST) and fractional revival (FR). We will do so in the
framework of algebraic graph theory.
We describe classifications of several graph properties concerning the so-called continuous quantum walks by just using the spectral information of the graphs. Even though there are expressions
for periodicity, PST and FR on graphs in terms of its eigenvalues, it remains a difficult task to find
(new examples of) graphs allowing the discussed properties. We will therefore pass from general
graphs to graphs belonging to association schemes. For these types of graphs there exists a characterisation that yields new examples of graphs admitting PST and FR. In this framework, we are able
to find large families of graphs on which PST and FR occurs. In particular, the results on association
schemes will apply to the well-known distance regular graphs

Datum prijs	30 jun 2022
Originele taal	English