Classical random walks have found applications ranging from Brownian motion, diffusion of particles, designing computer algorithms, and even in the study of biological systems.Random walks can be used to explain the process of diffusion in the driftdiffusion model charge carrier motion in semiconductors.In the classical treatment of transport phenomena, the charge carrier transport can be characterized by a well defined diffusion coefficient D. Recently, there have been numerous proposals to make electronic and logic devices based on the spin property of electron like the Datta-Das Spin Field Effect Transistor.In such devices, the transport is non-classical (ballistic).This document examines the possibility of defining a spin diffusion coefficient for nonclassical transport, using the study of quantum random walks as a starting point.In addition, this document also analyzes the effect of uniform noise on the behavior of quantum random walks and driving the quantum walk into classical regime.i Boolchand for his constant motivation and support throughout my stay at the University of Cincinnati and his valuable suggestions.I would like to thank Dr. Altan Ferendeci, who was on my Master's defense committee, for his suggestions to improve the quality of this work.I extend my thanks to Dr. Rohana Wijewardhana in the Physics department of University of Cincinnati for allowing me to attend his lectures.I thank my lab mates Maitreya Datta and Nikhil Bhandari for their encouragement and the stimulating environment in the lab.Special thanks are in order to my multitude of friends, those in Cincinnati for making my stay here enjoyable and those elsewhere for their support, encouragement and belief in
Classical random walks have found applications ranging from Brownian motion, diffusion of particles, designing computer algorithms, and even in the study of biological systems.Random walks can be used to explain the process of diffusion in the driftdiffusion model charge carrier motion in semiconductors.In the classical treatment of transport phenomena, the charge carrier transport can be characterized by a well defined diffusion coefficient D. Recently, there have been numerous proposals to make electronic and logic devices based on the spin property of electron like the Datta-Das Spin Field Effect Transistor.In such devices, the transport is non-classical (ballistic).This document examines the possibility of defining a spin diffusion coefficient for nonclassical transport, using the study of quantum random walks as a starting point.In addition, this document also analyzes the effect of uniform noise on the behavior of quantum random walks and driving the quantum walk into classical regime.i Boolchand for his constant motivation and support throughout my stay at the University of Cincinnati and his valuable suggestions.I would like to thank Dr. Altan Ferendeci, who was on my Master's defense committee, for his suggestions to improve the quality of this work.I extend my thanks to Dr. Rohana Wijewardhana in the Physics department of University of Cincinnati for allowing me to attend his lectures.I thank my lab mates Maitreya Datta and Nikhil Bhandari for their encouragement and the stimulating environment in the lab.Special thanks are in order to my multitude of friends, those in Cincinnati for making my stay here enjoyable and those elsewhere for their support, encouragement and belief in
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.
We investigate the quantum walk on the line when decoherence is introduced through simultaneous measurements of the chirality and the particle position. This decoherence drives the system to a classical diffusive behavior, for which we have obtained analytical expressions for its diffusion coefficient. We show that this coefficient is proportional to the variance of the initially localized quantum random walker just before the first measurement. Through an association of the measurement processes to the collisions in Brownian motion, we show that this treatment provides a smooth connection between the pure classical and quantum limits.
Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms, and quantum simulation of condensed-matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speedup in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct prediffusive and diffusive behaviors at different timescales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speedup, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speedup implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.
Quantum versions of random walks have diverse applications that are motivating experimental implementations as well as theoretical studies. However, the main impetus behind this interest is their use in quantum algorithms, which have always employed the quantum walk in the form of a program running on a quantum computer. Recent results showing that quantum walks are "universal for quantum computation" relate entirely to algorithms, and do not imply that a physical quantum walk could provide a new architecture for quantum computers. Nonetheless, quantum walks used to model transport phenomena in spin chains and biomolecules broaden their scope well beyond algorithms, and reopen the question of when a physical implementation might provide useful computational outputs. In this article we determine the conditions under which a physical quantum walk experiment could provide useful results beyond the reach of classical computation.
We show that particle transport, as characterized by the equilibrium mean square displacement, in a uniform, quantum multibaker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semiclassical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We expect that, for a large class of 1D quantum random walks similar to the quantum multibaker, a sufficient condition for diffusion in the semiclassical limit is classically chaotic dynamics in each cell. The systems described generalize known quantum random walks and may have applications for quantum computation.
Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.