Over the last two decades quantum walks have proven to be a very useful concept in quantum information processing, quantum communication and quantum simulations. The presented thesis reviews our contribution to this field of research. We provide a thorough introduction to methods used for the study of quantum walks and their potential applications. To keep the presentation intelligible we focus on specific examples. In particular, we analyze in detail the properties of homogeneous quantum walks on a one-dimensional lattice with two and three internal states. The application of the obtained results is illustrated on the examples persistence of unvisited sites and quantum transport to an absorbing sink. We also discuss quantum walks on finite graphs and their applications to quantum search and perfect state transfer. Extension of the presented results, which can be found in enclosed published papers, is discussed in detail.
Over the last two decades quantum walks have proven to be a very useful concept in quantum information processing, quantum communication and quantum simulations. The presented thesis reviews our contribution to this field of research. We provide a thorough introduction to methods used for the study of quantum walks and their potential applications. To keep the presentation intelligible we focus on specific examples. In particular, we analyze in detail the properties of homogeneous quantum walks on a one-dimensional lattice with two and three internal states. The application of the obtained results is illustrated on the examples persistence of unvisited sites and quantum transport to an absorbing sink. We also discuss quantum walks on finite graphs and their applications to quantum search and perfect state transfer. Extension of the presented results, which can be found in enclosed published papers, is discussed in detail.
In this thesis we present a theoretical study of quantum walks, with a particular focus on the development of viable schemes concerned with their physical realization. Ever since their introduction over a decade ago, quantum walks have been extensively explored for their non-intuitive dynamics which may hold the key to a new generation of quantum algorithms. This growing interest in the theoretical applications of quantum walks has been paralleled by a flurry of research into a more practical problem: how does one physically implement a quantum walk in the laboratory? We begin this thesis by first presenting an overview of the quantum walk theory, including some of its algorithmic applications. This is then followed by a comprehensive survey of numerous proposals for a physical implementation of quantum walks, underpinned by a wide range of quantum, classical and hybrid technologies. This review consequently highlights what has so far remained a major challenge for the quantum walk enthusiasts; a physical realization that is experimentally viable whilst remaining readily scalable and not limited to problems with specific connectivity criteria. It is precisely this challenge that we seek to examine in the remaining parts of this thesis.
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.
Quantum computing aims to harness and exploit the quantum mechanical phenomena such as superposition, entanglement and contextuality in order to encode and process information. In this context, quantum walks, which has been suggested as the quantum counterpart of classical random walks, is an emerging topic in quantum computing that provides powerful techniques for developing new quantum algorithms, quantum simulation and quantum state transfer. This thesis intends to investigate the properties of quantum walks which may potentially promote further work in such techniques in quantum computation. We first propose a novel method for transferring arbitrary unknown qubit state between two points in space with quantum walk architecture. We determine the cases providing perfect state transfers over both finite and infinite lattices with different boundary conditions and we introduce recovery operators assisting the transfer process. Next, by modeling the incoherent and coherent transport with classical random walks and quantum walks, respectively, we calculate the transport efficiencies over an explosive percolation lattice. We show that the minimal correlation between discrete clusters leads to maximal localizations which originating from random scatterings. These localization effects, however, are rather small when compared to the supportive effect of the abrupt growth of the largest cluster on transport efficiency, which eventually allows us to obtain more efficient transports in case of minimal correlations. We support our results with further calculations on whether the eigenstates of the systems we study are localized or not. Lastly, we turn our attention to the spreading dynamics and the coin-position entanglement for two-dimensional quantum walks under an artificial magnetic fields by introducing Peierls phases to the system. In particular, we show that the spreading of the quantum walk is diffusive rather than ballistic when the ratio of the magnetic flux through unit cell - where the walk takes place - to the flux quantum is an irrational number. On the contrary, the walk recovers its original ballistic behavior when this ratio is chosen to be a rational number. Furthermore, we demonstrate that coin-position entanglement is nearly maximum under an artificial magnetic field, even for large number of steps.
The classical random walk formalism plays an important role in a wide range of applications. Its quantum counterpart, the quantum walk, is proposed as an important theoretical model for quantum computing. By exploiting quantum effects such as superposition, interference, and entanglement, quantum walks and their variations have been extensively studied for achieving computing power beyond that of classical computing and have been broadly used in designing quantum algorithms for algebraic and optimization problems, graph and network analysis, and quantum Hamiltonian and biochemical process simulations. Moreover, quantum walk models have been proven capable of universal quantum computation. Unlike conventional quantum circuit models, quantum walks provide a feasible path for implementing application-specific quantum computing, particularly in the noisy intermediate-scale quantum era. Recently, remarkable progress has been achieved in implementing a wide variety of quantum walks and quantum walk applications, which demonstrates the great potential of quantum walks. In this review, we provide a thorough summary of quantum walks and quantum walk computing, including theories and characteristics, physical implementations, and applications. We also discuss the challenges facing quantum walk computing, which aims to realize a practical quantum computer in the near future.
Abstract Quantum walk is the quantum version of classical random walk, which has shown great advantage over classical algorithms. In this article, main discoveries and breakthroughs along with several applications in the last two decades are reviewed and discussed. Reducing hitting time and diminishing mixing time, as two hot research fields, are addressed. Possibility of universal computation with more generic case of quantum walk is also mentioned. In the future, we can expect new methods and models being feasible ways to approach the theoretical lower bounds of related problems, as well as more efficient applications on specific scene.
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.