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© 2013

Quantum Walks and Search Algorithms

Textbook

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

  1. 1., Department of Computer ScienceNational Laboratory of Scientific ComputPetrópolisBrazil

About the authors

Dr. Renato Portugal is Researcher in the Department of Computer Science at the National Laboratory for Scientific Computing (LNCC). His past positions include Visiting Professor in the Department of Applied Mathematics and the Symbolic Computation Group at the University of Waterloo, Visiting Professor in the Department of Physics at Queen’s University of Kingston, and Researcher at the Brazilian Center for Research in Physics. He received his D.Sc. at the Centro Brasileiro de Pesquisas Fisicas, CBPF, Brazil. He has published 40 articles in Scientific Journals, 3 books, and over 30 papers in refereed proceedings. He has developed 7 software packages, including his latest: The Invar Package in 2007. He was General Chair of the Workshop-School of Quantum Information and Computation (WECIQ 2010), and Chair of the Programme Committee for the Workshop-School of Quantum Information and Computation (WECIQ 2006).

Bibliographic information

Industry Sectors
Electronics
Energy, Utilities & Environment
IT & Software

Reviews

From the reviews:

“The reviewed book is a pedagogically oriented survey of the main results regarding quantum walks and quantum search algorithms. … The book is nicely written, the concepts are introduced naturally, and many meaningful connections between them are highlighted. The author proposes a series of exercises that help the reader get some working experience with the presented concepts, facilitating a better understanding. Each chapter ends with a discussion of further references, pointing the reader to major results on the topics presented in the respective chapter.” (Florin Manea, zbMATH, Vol. 1275, 2014)