Application of graph theory in quantum computer science

TL;DR

This paper explores the extension of continuous-time quantum walks to directed graphs and investigates quantum speed-up in complex graph structures, advancing quantum algorithms and graph theory applications.

Contribution

It introduces methods for implementing quantum walks on directed graphs and analyzes quantum speed-up on complex graph models, expanding the scope of quantum walk applications.

Findings

Quantum walks can be constructed on directed graphs with preserved propagation.

Quantum speed-up observed in complex graph structures.

Extended understanding of quantum walk behavior on realistic graph models.

Abstract

In this dissertation we demonstrate that the continuous-time quantum walk models remain powerful for nontrivial graph structures. We consider two aspects of this problem. First, it is known that the standard Continuous-Time Quantum Walk (CTQW), proposed by Childs and Goldstone, can propagate quickly on the infinite path graph. However, the Schr\"odinger equation requires the Hamiltonian to be symmetric, and thus only undirected graphs can be implemented. In this thesis, we address the question, whether it is possible to construct a continuous-time quantum walk on general directed graphs, preserving its propagation properties. Secondly, the quantum spatial search defined through CTQW has been proven to work well on various undirected graphs. However, most of these graphs have very simple structures. The most advanced results concerned the Erd\H{o}s-R\'enyi model of random graphs, which is the most popular but not realistic random graph model, and Barab\'asi-Albert random graphs, for which full quadratic speed-up was not confirmed. In the scope of this aspect we analyze, whether quantum speed-up is observed for complicated graph structures as well.