Abstract. The adjacency matrix of a graph [Formula: see text] is the Hamiltonian for a continuous-time quantum walk on the vertices of [Formula: see text]. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues—what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.
Abstract. The adjacency matrix of a graph [Formula: see text] is the Hamiltonian for a continuous-time quantum walk on the vertices of [Formula: see text]. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues—what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.
Quantum computing is believed to provide many advantages over traditional computing, particularly considering the speed at which computations can be performed. One of the challenges that needs to be resolved in order to construct a quantum computer is the transmission of information from one part of the computer to another. Quantum walks, the quantum analogues of classical random walks, provide one potential method for resolving this challenge. \n \nIn this thesis, we use techniques from algebraic graph theory and number theory to analyze the mathematical model for continuous time quantum walks on graphs. For the continuous time quantum walk model, we define a transition operator, which is a function of a Hamiltonian. We focus on the cases where the adjacency matrix or the Laplacian of a graph act as the Hamiltionian. We mainly consider quantum walks on paths as a model for spin chains, which are the underlying basis of a quantum communication protocol. \n \nFor communication to be efficient, we desire states to be transferred with high fidelity, a measure of the amount of similarity between the transmitted state and the received state. At the maximum fidelity of 1, we say we have achieved perfect state transfer. Examples of perfect state transfer are relatively rare, so the concept of pretty good state transfer was introduced as a natural relaxation, which exists if fidelities arbitrarily close to 1 are obtained. \n \nOur first main result is to characterize pretty good state transfer on paths. Previously, pretty good state transfer on paths was considered mainly for the end vertices, though results for both models indicated that if there was pretty good state transfer between the end vertices, then there was pretty good state transfer between internal vertices equidistant from each end. We complete the characterization by demonstrating, for the adjacency matrix model, a family of paths where pretty good state transfer exists between internal vertices but not between end vertices, and verifying that no other example exists. For the Laplacian model, we show that there are no paths with pretty good state transfer between internal vertices but not between the end vertices. \n \nOur second main result considers initial states involving multiple vertices. Under the adjacency matrix model, we provide necessary and sufficient conditions for pretty good state transfer in a particular family of paths in terms of the eigenvalue support of the initial state. We also discuss recent results on fractional revival, which is another form of multiple qubit state transfer.
A continuous time quantum walk consists of a particle moving among the vertices of a graph G. Its movement is governed by the structure of the graph. More formally, the adjacency matrix A is the Hamiltonian that determines the movement of our particle. Quantum walks have found a number of algorithmic applications, including unstructured search, element distinctness and Boolean formula evaluation. We will examine the properties of periodicity and state transfer. In particular, we will prove a result of the author along with Godsil, Kirkland and Severini, which states that pretty good state transfer occurs in a path of length n if and only if the n+1 is a power of two, a prime, or twice a prime. We will then examine the property of strong cospectrality, a necessary condition for pretty good state transfer from u to v. \n \nWe will then consider quantum walks involving more than one particle. In addition to moving around the graph, these particles interact when they encounter one another. Varying the nature of the interaction term gives rise to a range of different behaviours. We will introduce two graph invariants, one using a continuous-time multi-particle quantum walk, and the other using a discrete-time quantum walk. Using cellular algebras, we will prove several results which characterize the strength of these two graph invariants. \n \nLet A be an association scheme of n × n matrices. Then, any element of A can act on the space of n × n matrices by left multiplication, right multiplication, and Schur multiplication. The set containing these three linear mappings for all elements of A generates an algebra. This is an example of a Jaeger algebra. Although these algebras were initially developed by Francois Jaeger in the context of spin models and knot invariants, they prove to be useful in describing multi-particle walks as well. We will focus on triply-regular association schemes, proving several new results regarding the representation of their Jaeger algebras. As an example, we present the simple modules of a Jaeger algebra for the 4-cube.
Abstract We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon.
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. \n \nThere 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. \n \nWe 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. \n \nNext 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. \n \nFinally 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.
In this paper, we introduce a discrete quantum walk model called bipartite walks. Bipartite walks include many known discrete quantum walk models, like Grover’s walks, vertex-face walks. For the transition matrix of a quantum walk, there is a Hamiltonian associated with it. We will study the Hamiltonians of the bipartite walks. Let S be a skew-symmetric matrix. We are mainly interested in the Hamiltonians of the form iS. We show that the Hamiltonian can be written as iS if and only if the adjacency matrix of the bipartite graph is invertible. We show that Grover’s walks and vertex-face walks are special cases of bipartite walks. Via the Hamiltonians, phenomena of bipartite walks lead to phenomena of continuous walks. We show in detail how we use bipartite walks on paths to construct universal perfect state transfer in continuous walks.
In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.