This thesis extends results about periodicity and perfect state transfer \nto oriented graphs. We prove that if a vertex a is periodic, then elements of \nthe eigenvalue support lie in Z √∆ for some squarefree negative integer \n∆. We find an infinite family of orientations of the complete graph that are \nperiodic. We find an example of a graph with both perfect state transfer \nand periodicity that is not periodic at an integer multiple of the period, and \nwe prove and use Gelfond-Schneider Theorem to show that every oriented \ngraph with perfect state transfer between two vertices will have both vertices \nperiodic. We find a complete characterization of when perfect state transfer \ncan occur in oriented graphs, and find a new example of a graph where one \nvertex has perfect state transfer to multiple other vertices.
This thesis extends results about periodicity and perfect state transfer \nto oriented graphs. We prove that if a vertex a is periodic, then elements of \nthe eigenvalue support lie in Z √∆ for some squarefree negative integer \n∆. We find an infinite family of orientations of the complete graph that are \nperiodic. We find an example of a graph with both perfect state transfer \nand periodicity that is not periodic at an integer multiple of the period, and \nwe prove and use Gelfond-Schneider Theorem to show that every oriented \ngraph with perfect state transfer between two vertices will have both vertices \nperiodic. We find a complete characterization of when perfect state transfer \ncan occur in oriented graphs, and find a new example of a graph where one \nvertex has perfect state transfer to multiple other vertices.
The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor Δ, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order n, we derive an explicit formula for every Fourier eigenvalue without assuming that n/Δ is coprime to Δ. We prove that PST occurs only for Δ∈{3,4,8} and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly 1, 2, and 3. More generally, when λj=Dηj with ηj∈Z, congruences satisfied by the integers ηj determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.
Given a graph with Hermitian adjacency matrix H, perfect state transfer occurs from vertex a to vertex b if the (b,a)-entry of the unitary matrix exp(−iHt) has unit magnitude for some time t. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are ±i), the oriented 3-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.
A discrete-time quantum walk is the quantum analogue of a Markov chain on a graph. We show that the evolution of a general discrete-time quantum walk that consists of two reflections satisfies a Chebyshev recurrence, under a projection. We apply this to study perfect state transfer in a model of discrete-time quantum walk whose transition matrix is given by two reflections, defined by the face and vertex incidence relations of a graph embedded in an orientable surface, proposed by Zhan [ J. Algebraic Combin. 53(4):1187–1213, 2020]. For this model, called the vertex-face walk, we prove results about perfect state transfer and periodicity and give infinite families of examples where these occur. In doing so, we bring together tools from algebraic and topological graph theory to analyze the evolution of this walk.
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.
For perfect state transfer (PST) on unweighted mixed graphs, we classify the normalized transfer times of complete PST classes. A finite set Λ⊂R/Z containing zero occurs at a nonstationary periodic vertex if and only if cos(2π(x−y))∈Q for all x,y∈Λ. Every admissible set has a connected oriented realization. We also classify the possible return phases of oriented realizations at the minimum vertex period. Transfers at rational multiples of the common minimum vertex period partition a complete class into sets of size at most six, or at most three in an oriented graph with return phase −1; both bounds are sharp. We construct complete classes of every finite size, including classes in which all transfers between distinct vertices occur at irrational multiples of the period and no switching automorphism maps a class vertex to a distinct class vertex. We also characterize simultaneous realization in connected oriented graphs with prescribed relative minimum vertex periods and return phases. The proof combines an imaginary quadratic field restriction with an unweighted construction that selects the complete target set.
The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices u and v are adjacent if gcd(u−v,n)=1. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs.