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Quantum Walks and Search Algorithms

  • Renato Portugal

Part of the Quantum Science and Technology book series (QST)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Renato Portugal
    Pages 1-2
  3. Renato Portugal
    Pages 3-16
  4. Renato Portugal
    Pages 17-37
  5. Renato Portugal
    Pages 39-63
  6. Renato Portugal
    Pages 65-83
  7. Renato Portugal
    Pages 85-120
  8. Renato Portugal
    Pages 121-143
  9. Renato Portugal
    Pages 145-163
  10. Renato Portugal
    Pages 165-193
  11. Back Matter
    Pages 195-222

About this book

Introduction

This book addresses an interesting area of quantum computation called quantum walks, which play an important role in building quantum algorithms, in particular search algorithms. Quantum walks are the quantum analogue of classical random walks.

It is known that quantum computers have great power for searching unsorted databases. This power extends to many kinds of searches, particularly to the problem of finding a specific location in a spatial layout, which can be modeled by a graph. The goal is to find a specific node knowing that the particle uses the edges to jump from one node to the next.

This book is self-contained with main topics that include:

  • Grover's algorithm, describing its geometrical interpretation and evolution by means of the spectral decomposition of the evolution operater
  • Analytical solutions of quantum walks on important graphs like line, cycles, two-dimensional lattices, and hypercubes using Fourier transforms
  • Quantum walks on generic graphs, describing methods to calculate the limiting distribution and mixing time
  • Spatial search algorithms, with emphasis on the abstract search algorithm (the two-dimensional lattice is used as an example)
  • Szedgedy's quantum-walk model and a natural definition of quantum hitting time (the complete graph is used as an example)

The reader will benefit from the pedagogical aspects of the book, learning faster and with more ease than would be possible from the primary research literature. Exercises and references further deepen the reader's understanding, and guidelines for the use of computer programs to simulate the evolution of quantum walks are also provided.

Keywords

Abstract Search Algorithms Continuous-time Algorithms Discrete-time Quantum Walks Grover’s Algorithm Quantum Algorithms Quantum Hitting Times Quantum Markov Chains Quantum Mixing Times Quantum Search Algorithms Quantum Walks on Graphs Szegedy’s Quantum-walk Model

Authors and affiliations

  • Renato Portugal
    • 1
  1. 1., Department of Computer ScienceNational Laboratory of Scientific ComputPetrópolisBrazil

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-6336-8
  • Copyright Information Springer Science+Business Media New York 2013
  • Publisher Name Springer, New York, NY
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-1-4614-6335-1
  • Online ISBN 978-1-4614-6336-8
  • Buy this book on publisher's site