Quantum Walks and the Renormalization Group Open Access

Falkner, Stefan (2014)

Permanent URL: https://etd.library.emory.edu/concern/etds/nc580n085?locale=en


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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