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CC BY V4.0, Open Access
Dissertation
Continuous-Time Quantum Walks on Windmill Graphs
Washington State University
Doctor of Philosophy (PhD), Washington State University
2023
DOI:
https://doi.org/10.7273/000006322
Abstract
Let $A$ be the adjacency matrix of a graph. We will associate this graph with a continuous-time quantum walk by using a transition matrix $U(t) = e^{itA}$. This allows us to create another matrix $\hat{M}$ which is time-independent. $\hat{M}$ gives us some measure of average probability values and long-term behavior of a continuous time quantum walk and is called the average mixing matrix. This has been studied extensively on trees and other graphs with distinct eigenvalues.
Our work focuses on graphs which have repeated eigenvalues. We give our own proof for why Star graphs have full rank, and we present an entry-wise recursive way to express the eigenvectors of Path graphs. We utilize this result to prove that the rank of Dutch Windmills with 3 or more $P_m$ blades is $\left\lceil\frac{m}{2}\right\rceil b + 1$. We go on to prove that French Windmills (appending $K_m$ instead) have full rank and that Kulli Cycle Windmills (appending $C_m$ alternate between full rank when $m$ is odd and $\frac{m}{2}b+1$ when $m$ is even.
Lastly we present some experimental results for Kulli Path Windmills which appear to mirror the behavior of Dutch Windmills. Separately, we demonstrate that when adding blades of different lengths onto an initially regular Dutch Windmill, it does not always add the rank of the blade sub-matrix as might be expected.
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Details
- Title
- Continuous-Time Quantum Walks on Windmill Graphs
- Creators
- Paula Rachael Kimmerling
- Contributors
- Judith McDonald (Advisor)Michael Tsatsomeros (Committee Member)Matthew Hudelson (Committee Member)Sergey Lapin (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Department of Mathematics and Statistics
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 100
- Identifiers
- 99901086723401842
- Language
- English
- Resource Type
- Dissertation