Discrete-Time Quantum Walk - Dynamics and Applications

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.Comment: 199 pages, 52 figures, Thesis completed during 2009 at University of Waterloo (IQC), V2 : Index of figures has been made compac

Measuring graph similarity through continuous-time quantum walks and the quantum Jensen-Shannon divergence

In this paper we propose a quantum algorithm to measure the similarity between a pair of unattributed graphs. We design an experiment where the two graphs are merged by establishing a complete set of connections between their nodes and the resulting structure is probed through the evolution of continuous-time quantum walks. In order to analyze the behavior of the walks without causing wave function collapse, we base our analysis on the recently introduced quantum Jensen-Shannon divergence. In particular, we show that the divergence between the evolution of two suitably initialized quantum walks over this structure is maximum when the original pair of graphs is isomorphic. We also prove that under special conditions the divergence is minimum when the sets of eigenvalues of the Hamiltonians associated with the two original graphs have an empty intersection

Two Lectures on Technicolor

These two lectures on technicolor and extended technicolor (ETC) were presented at l'Ecole de GIF at LAPP, Annecy-le-Vieux, France, in September 2001. In Lecture I, the motivation and structure of this theory of dynamical breaking of electroweak and flavor symmetries is summarized. The main phenomenological obstacles to this picture--flavor-changing neutral currents, precision electroweak measurements, and the large top-quark mass--are reviewed. Then, their proposed resolutions--walking technicolor and topcolor-assisted technicolor are discussed. In Lecture II, a scenario for CP violation is presented based on vacuum alignment for technifermions and quarks. It has the novel feature of CP--violating phases that are rational multiples of pi to better than one part in 10^{10} without fine-tuning of parameters. The scheme thereby avoids light axions and a massless up quark. The mixing of neutral mesons, the mechanism of top--quark mass generation, and the CP--violating parameters epsilon and sin(2 beta) strongly constrain the form of ETC--generated quark mass matrices.Comment: 60 pages, LaTex, with two postscript figure