The quantum walk is a method for conceptualizing and designing quantum computing algorithms and it comes in two forms: the continuous-time and discrete-time quantum walk. The thesis is organized into three parts, each of which looks to develop the concept and uses of the quantum walk. The first part is the theory of the quantum walk. This includes definitions and considerations for the various incarnations of the discrete-time quantum walk and a discussion on the general method for connecting the continuous-time and discrete-time versions. As a result, it is shown that most versions of the discrete-time quantum walk can be put into a general form and this can be used to simulate any continuous-time quantum walk. The second part uses these results for a hypothetical application. The application presented is a search algorithm that appears to scale in the time for completion independent of the size of the search space. This behavior is then elaborated upon and shown to have general qualitative agreement with simulations to within the approximations that are made. The third part introduces a method of implementation. Given a universal quantum computer, the method is discussed and shown to simulate an arbitrary discrete-time quantum walk. Some of the benefits of this method are that half the unitary evolution can be achieved without the use of any gates and there may be some possibility for error detection. The three parts combined suggest a possible experiment, given a quantum computing scheme of sufficient robustness.
The quantum walk is a method for conceptualizing and designing quantum computing algorithms and it comes in two forms: the continuous-time and discrete-time quantum walk. The thesis is organized into three parts, each of which looks to develop the concept and uses of the quantum walk. The first part is the theory of the quantum walk. This includes definitions and considerations for the various incarnations of the discrete-time quantum walk and a discussion on the general method for connecting the continuous-time and discrete-time versions. As a result, it is shown that most versions of the discrete-time quantum walk can be put into a general form and this can be used to simulate any continuous-time quantum walk. The second part uses these results for a hypothetical application. The application presented is a search algorithm that appears to scale in the time for completion independent of the size of the search space. This behavior is then elaborated upon and shown to have general qualitative agreement with simulations to within the approximations that are made. The third part introduces a method of implementation. Given a universal quantum computer, the method is discussed and shown to simulate an arbitrary discrete-time quantum walk. Some of the benefits of this method are that half the unitary evolution can be achieved without the use of any gates and there may be some possibility for error detection. The three parts combined suggest a possible experiment, given a quantum computing scheme of sufficient robustness.
Random walks are a powerful tool for the efficient implementation of algorithms in classical computation. Their quantum-mechanical analogues, called quantum walks, hold similar promise. Quantum walks provide a model of quantum computation that has recently been shown to be equivalent in power to the standard circuit model. As in the classical case, quantum walks take place on graphs and can undergo discrete or continuous evolution, though quantum evolution is unitary and therefore deterministic until a measurement is made. This thesis considers the usefulness of continuous-time quantum walks to quantum computation from the perspectives of both their fundamental power under various formulations, and their applicability in practical experiments. In one extant scheme, logical gates are effected by scattering processes. The results of an exhaustive search for single-qubit operations in this model are presented. It is shown that the number of distinct operations increases exponentially with the number of vertices in the scattering graph. A catalogue of all graphs on up to nine vertices that implement single-qubit unitaries at a specific set of momenta is included in an appendix. I develop a novel scheme for universal quantum computation called the discontinuous quantum walk, in which a continuous-time quantum walker takes discrete steps of evolution via perfect quantum state transfer through small `widget' graphs. The discontinuous quantum-walk scheme requires an exponentially sized graph, as do prior discrete and continuous schemes. To eliminate the inefficient vertex resource requirement, a computation scheme based on multiple discontinuous walkers is presented. In this model, n interacting walkers inhabiting a graph with 2n vertices can implement an arbitrary quantum computation on an input of length n, an exponential savings over previous universal quantum walk schemes. This is the first quantum walk scheme that allows for the application of quantum error correction. The many-particle quantum walk can be viewed as a single quantum walk undergoing perfect state transfer on a larger weighted graph, obtained via equitable partitioning. I extend this formalism to non-simple graphs. Examples of the application of equitable partitioning to the analysis of quantum walks and many-particle quantum systems are discussed.
The field of quantum information applies the concepts of quantum physics to problems in computer science, and shows great potential for allowing efficient computation. \nIn this thesis I concentrate on a particular quantum information theoretic tool known as the quantum walk. There are two widely studied versions of the quantum walk, the continuous time walk, and the discrete time walk. The discrete time walk is particularly amenable to investigation using numerical methods, from which most of the results in this thesis are derived, and is the main focus of the work presented. Two aspects of the discrete time walk are investigated: their transport properties \nand their interpretation as quantum computers. I investigated the transport properties in two ways, by looking for a particular type of transport known as perfect \nstate transfer, and examining the transport properties of a new type of coin operator. \nThe search for perfect state transfer concentrated on modifications of small cycles. I found that perfect state transfer is rare for the choices of coin operators \ntested. The structures tested for perfect state transfer were based on cycles, and it appears that the type of modification has more of an effect than the size of the \ncycle. This makes intuitive sense, as the modifications found to lead to walks exhibiting perfect state transfer affected only the initial and target node of the cycle. \nI then investigated a new type of coin operator which does not allow amplitude to return to the node it has come from. This effectively simulates a dimer. Using the general form of this type of operator and random variables for each parameter, I found that the expected distance of the walker from the origin, and standard deviation, were independent of the initial condition. The second half of the thesis concentrates on computational applications of quantum \nwalks using the language acceptance model. I first note their equivalence to a type of quantum automaton known as the QFA-WOM, and this provides an intuitive understanding of the role of the WOM. I then use a more direct construction to show that they can accept a range of formal languages. Using this construction allows us to use superpositions of words as inputs, and the insights provided by investigating these suggest a new way of approaching the problem of quantum state discrimination.
This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. \n \nIn the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. \n \nWe investigate symmetry operations and environmental effects on dynamics of the walk on a line and an n−cycle highlighting the interplay between noise and topology. \n \nUsing the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.
This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. In the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. We investigate symmetry operations and environmental effects on dynamics of the walk on a line and an n-cycle highlighting the interplay between noise and topology. Using the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.
Quantum computation is a new computational paradigm which can provide fundamentally faster computation than in the classical regime. This is dependent on finding efficient quantum algorithms for problems of practical interest. One of the most successful tools in developing new quantum algorithms is the quantum walk. In this thesis, we explore two applications of the discrete time quantum walk. In addition, we introduce an experimental scheme for generating cluster states, a universal resource for quantum computation. We give an explicit construction which provides a link between the circuit model of quantum computation, and a graph structure on which the discrete time quantum walk traverses, performing the same computation. We implement a universal gate set, proving the discrete time quantum walk is universal for quantum computation, thus confirming any quantum algorithm can be recast as a quantum walk algorithm. In addition, we study factors affecting the efficiency of the quantum walk search algorithm. Although there is a strong dependence on the spatial dimension of the structure being searched, we find secondary dependencies on other factors including the connectivity and disorder (symmetry). Fairly intuitively, as the connectivity increases, the efficiency of the algorithm increases, as the walker can coalesce on the marked state with higher probability in a quicker time. In addition, we find as disorder in the system increases, the algorithm can maintain the quantum speed up for a certain level of disorder before gradually reverting to the classical run time. Finally, we give an abstract scheme for generating cluster states. We see a linear scaling, better than many schemes, as doubling the size of the generating grid in our scheme produces a cluster state which is double the depth. Our scheme is able to create other interesting topologies of entangled states, including the unit cell for topological error correcting schemes.
Quantum computing is a developing field that applies the properties of quantum mechanics to develop solutions to complex problems that the most powerful classical computers cannot solve or cannot solve efficiently. Solving difficult tasks requires sophisticated and efficient algorithms that can utilize stochastic processes as the primary techniques. The typical models of these stochastic processes include discrete-time random walks and continuous ones. Quantum walks represent a quantum extension of discrete random walks, whose demonstrated utility in algorithm design has motivated signi cant interest in quantum computational research. Therefore, the purpose of this thesis is to explore the potential of discrete-time coined quantum walks using the Python package with high-performance computing (HPC) compared to classical walks by conducting a literature review. To supplement this research, some simulations are implemented to analyze the spreading behavior of coined quantum walks, compare them with classical walks, and demonstrate their application in search tasks, such as binary hypercube search algorithm. These simulations aim to clarify the extent to which quantum walks can be effectively modeled and applied in computational contexts. The results gained show that coined quantum walks exhibit faster spreading than classical random walks, which contributes to a deeper understanding of potential applications of quantum walks, such as quantum simulation, in the future.