Uniform Mixing on Cayley Graphs over Z_3^d

This thesis investigates uniform mixing on Cayley graphs over Z_3^d. We apply Mullin's results on Hamming quotients, and characterize the 2(d+2)-regular connected Cayley graphs over Z_3^d that admit uniform mixing at time 2pi/9. We generalize Chan's construction on the Hamming scheme H(d,2) to the scheme H(d,3), and find some distance graphs of the Hamming graph H(d,3) that admit uniform mixing at time 2pi/3^k for any k≥2. To restrict the mixing time, we derive a sufficient and necessary condition for uniform mixing to occur on a Cayley graph over Z_3^d at a given time. Using this, we obtain three results. First, we give a lower bound of the valency of a Cayley graph over Z_3^d that could admit uniform mixing at some time. Next, we prove that no Hamming quotient H(d,3)/ admits uniform mixing at time earlier than 2pi/9. Finally, we explore the connected Cayley graphs over Z_3^3 with connected complements, and show that five complementary graphs admit uniform mixing with earliest mixing time 2pi/9

Uniform Mixing of Quantum Walks and Association Schemes

In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. There are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. We begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. Next we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. Finally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups

Some results on uniform mixing on abelian Cayley graphs

In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. Among them, uniform mixing provides a uniform probability distribution of quantum information over time which attracts a special attention. However, there are only a few known examples of graphs which admit uniform mixing. In this paper, a characterization of abelian Cayley graphs having uniform mixing is presented. Some concrete constructions of such graphs are provided. Specifically, for cubelike graphs, it is shown that the Cayley graph ${\rm Cay}(\mathbb{F}_2^{2k};S)$ has uniform mixing if the characteristic function of $S$ is bent. Moreover, a difference-balanced property of the eigenvalues of an abelian Cayley graph having uniform mixing is established. Furthermore, it is proved that an integral abelian Cayley graph exhibits uniform mixing if and only if the underlying group is one of the groups: $\mathbb{Z}_2^d, \mathbb{Z}_3^d$, $\mathbb{Z}_4^d$ or $\mathbb{Z}_2^{r}\otimes \mathbb{Z}_4^d$ for some integers $r \geq 1, d\geq 1$. Thus the classification of integral abelian Cayley graphs having uniform mixing is completed.Comment: 33 page

Sedentariness in quantum walks

We present a relaxation of the concept of a sedentary family of graphs introduced by Godsil [Linear Algebra Appl. 614:356-375, 2021] and provide sufficient conditions for a given vertex in a graph to exhibit sedentariness. We show that a vertex with at least two twins (vertices that share the same neighbours) is sedentary. We also prove that there are infinitely many graphs containing strongly cospectral vertices that are sedentary, which reveals that, even though strong cospectrality is a necessary condition for pretty good state transfer, there are strongly cospectral vertices which resist high probability state transfer to other vertices. Moreover, we derive results about sedentariness in products of graphs which allow us to construct new sedentary families, such as Cartesian powers of complete graphs and stars.Comment: 26 pages, 3 figure

Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups

We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups, called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state transfer and fractional revival, and use partial spreads to construct graphs on extraspecial $2$-groups admitting these various phenomena. Lastly, we use a result of Ada Chan to show that there is no normal Cayley graph of an extraspecial group that admits instantaneous uniform mixing.Comment: 26 pages, minor correction and typ

Continuous-Time Quantum Walks on Windmill Graphs

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

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 $X$ is determined by a unitary matrix $U$, which acts on complex functions of the arcs of $X$. Generally speaking, $U$ 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 $X$, 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 $U$ is not defined as a function in the adjacency matrix of the graph $X$, we find exact spectral correspondence between $U$ and $X$. 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 $X$. 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 $U$. We show that knowing the spectral decomposition of $U$ 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 $U$. 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 $U$ has few eigenvalues are characterized. For instance, if $U$ has exactly three eigenvalues, then the vertex-face incidence structure is a symmetric $2$-design, and $U$ 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, $U$ is the transition matrix of a continuous quantum walk on an oriented graph