Quantum Walks and the Renormalization Group Öffentlichkeit
Falkner, Stefan (2014)
Published
Abstract
Like random walks, quantum walks are taking on a central role in
both describing physical transport phenomena, and establishing a
framework for quantum search algorithms. Unlike their classical
counterpart, quantum walks present a model for universal quantum
computation and can be used to simulate potentially any quantum
system. Despite recent theoretical and experimental advancements,
our understanding of quantum walks still lacks far behind that of
random walks as they exhibit a much broader spectrum of behaviors
awaiting categorization and context, even on simple lattices. Using
analytic and numerical methods, we explore dynamical properties of
quantum walks on self-similar networks. In particular, we study the
longtime asymptotic spreading on networks without translational
invariance. For one commonly studied quantum walk, we find a simple
relationship between quantum and classical dynamics suggesting a
deeper connection, yet to be understood. Furthermore, we show that
the parameters of quantum walks affect its dynamical properties on
these networks significantly. We also encounter a phenomenon called
localization where parts of the dynamics never move far from the
initial site. We contrast this behavior on regular lattices and a
fractal.
Table of Contents
1 Introduction
2 Discrete Time Walks and the Renormalization Group
2.1 Random Walks
2.1.1 Time evolution
2.1.2 Asymptotic Properties
2.2 Quantum Walks
2.2.1 Time Evolution
2.2.2 Asymptotic Properties
2.2.3 Coinless Quantum Walks
2.2.4 Physical Implementations
2.2.5 Quantum Walks and Quantum Search Algorithms
2.3 The Renormalization Group for Random Walks
2.4 The Renormalization Group for Quantum Walks
3 Quantum walks on self-similar networks
3.1 Self-similar networks
3.2 Direct numerical simulations
3.3 Exploring different coins
3.3.1 The Grover Coin
3.3.2 The Fourier Coin
3.3.3 An orthogonal Coin
3.4 Remarks
4 Localization in quantum walks without disorder
4.1 The one dimensional random walk
4.2 The three-state one dimensional quantum walk
4.2.1 Long time approximation
4.2.2 Intuitive explanation for the localization
4.3 Localization on the Dual Sierpinsky Gasket
5 Conclusions
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