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.
A continuous-time quantum walk on a graph is represented by the complex matrix , where is the adjacency matrix of and is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here, we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of -matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.
This paper investigates perfect state transfer (PST) in discrete-time quantum walks constructed via the shunt decomposition method. The walks are defined on a graph
G and its associated quotient graph
G/π, induced by an equitable partition
π. Through the shunt decomposition of
G, we derive an explicit relation between the shift operator of the parent graph
G and that of its quotient graph
G/π. We construct a reflection operator based on the characteristic matrix, which establishes a connection between the transition operator of the parent graph and that of its lower-dimensional quotient graph. We then prove that PST occurs on
G if and only if it occurs on
G/π. Furthermore, we express the unitary evolution operator of the quotient graph in terms of Chebyshev polynomials of the first kind, from which we derive explicit criteria for PST. As an application, we establish PST on the cycle graph
Cn at time
k=n/2, and lift the result to the parent graph
C2n via the equitable partition
π. We further show that if an equitable partition
π of
G induces a quotient isomorphic to
Kn↺, the complete digraph on
n vertices with a loop at every vertex, then PST occurs at step
k=n, and the walk is periodic at
k=2n. This framework is applied to two families of graphs, which are the complete bipartite digraph
Kn,n⇌ and the circulant graph
Circ(2n,S), where
S consists of all odd residues modulo
2n and
n=2s for some
s≥1, establishing PST in their respective line digraphs. Collectively, these results also answer the question posed by Godsil and Zhan concerning which shunt decompositions or embeddings of a graph admit PST.
Abstract. Pure states correspond to one-dimensional subspaces of [Formula: see text] represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state [Formula: see text] admits PST with another real pure state [Formula: see text], (ii) every connected graph admits PST between real pure states, and (iii) for any pair of real pure states [Formula: see text] and [Formula: see text] and for any time [Formula: see text], there exists a real symmetric matrix [Formula: see text] such that [Formula: see text] and [Formula: see text] admit PST relative to [Formula: see text] at time [Formula: see text]. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all [Formula: see text]-vertex simple unweighted graphs, the least minimum PST time between real pure states is attained by any join graph for the Laplacian case, while it is attained by the join of an empty graph and a complete graph of appropriate sizes for the adjacency case. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at none . [Formula: see text]
In this paper, we define directed partial join graphs with signed couplings and construct discrete-time quantum-walk transition operators for these graphs using the shunt-decomposition framework. The resulting transition operators apply to several important graph families, including complete graphs with loops, circulant partial joins, complete bipartite graphs, tensor powers of complete bipartite graphs, all with signed couplings. For each family, we identify the corresponding structure of the transition operator and derive necessary and sufficient conditions for periodicity and perfect state transfer (PST), when one of the two directed regular graphs admits PST. Based on these results, we identify two types of state transfer; internal PST, which occurs between vertices within the same graph, and coupling PST, which occurs between two components of join graphs. We further develop a double-cover construction for directed partial join graphs and derive conditions for periodicity and PST when the associated transition operators do not necessarily commute. Using this construction, we establish PST results for double covers of complete graphs with loops. In particular, we provide an example in which the complete graph
Kn does not exhibit PST for
n≥4, whereas a suitable partial join of
Kn exhibits PST when
n=2m,
m≥2. Hence, these results extend the class of graph families admitting PST in shunt-decomposition-based quantum walks and provide a unified framework for studying quantum state transfer in graph joins, products, and covers.
Quantum spin systems carry information in the form of quantum states, and the propagation of these quantum states is described by a quantum walk. In order to construct an operational quantum computer, the task of reliably transmitting quantum states from one part of a quantum computer to another must be accomplished. Quantum walks are known to be powerful tools in achieving this task. Pure states are quantum states represented by unit complex vectors. In this thesis, we use algebraic and combinatorial techniques to develop the theory of perfect state transfer on pure states in quantum walks, with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a spin network. We prove basic results about eigenvalue supports, periodicity, and strong cospectrality on pure states. We investigate a special type of strong cospectrality called m-strong cospectrality, and show that m-strongly cospectral pure states admit desirable quantum state transfer properties. Several characterisations of perfect state transfer are also given, one for general pure states, one for m-strongly pure states, and another one for real pure states. These characterisations give rise to algebraic and analytic properties of pure states admitting perfect state transfer. Moreover, we determine all complete graphs, complete bipartite graphs, cycles, and paths that admit perfect state transfer between m-strongly pure states. Constructions of infinite families of graphs admitting perfect state transfer are also given. We also characterise perfect state transfer between vertex states in joins and blowup graphs. We determine when the join operation preserves or induces perfect state transfer in the resulting graph. We also use blow-up graphs to construct new families of regular graphs with perfect state transfer. We use these two graph operations to demonstrate that graphs with perfect state transfer can be constructed from graphs that do not exhibit such a property. Finally, we investigate perfect state transfer on s-pair states – pure states that represent entanglement between two vertices. We establish combinatorial and spectral properties of s-pair states with perfect state transfer, and characterise its existence in complete graphs, complete bipartite graphs, paths and distance-regular graphs. Further, we construct infinite families of graphs with s-pair state transfer.