ALGB Seminar: Krystal Guo (Université Libre de Bruxelles)

Title: Quantum walks, state transfer and perturbations of graphs

Abstract: In this talk, we will give a brief introduction to the application of algebraic graph theory to an area of quantum computing. Due to recent developments in quantum computing, the study of quantum walks has emerged as important and timely. A quantum walk is a quantum process on a graph and is a computational primitive for quantum computation. Since quantum systems do not allow cloning of states, state transfer is seen as an alternative and systems which permit state transfer appear to be rare and interesting. There has been a considerable amount of success in approaching questions about continuous-time quantum walks with tools in linear algebra and algebraic graph theory and we will discuss one such instance during this talk.

A simple spectral question one might ask is: suppose we delete an edge from a graph, how does this change the eigenvalues? Generally speaking, this is a difficult question, so we can only answer it for a class of graphs with very few eigenvalues. We study a generalization of deleting an edge from a graph and give a general formula for the characteristic polynomial. For the class of strongly regular graphs, we give a more precise statement, which allows us to determine when perfect state transfer occurs in continuous quantum walks on strongly regular graphs.

No prior knowledge of quantum computing will be assumed.