Quantum Walks and Search Algorithms

This book explores quantum walks, which are important in building quantum algorithms. Coverage includes Grover's algorithm; Analytical solutions of quantum walks using Fourier transforms; Quantum walks on generic graphs; Spatial search algorithms and more.

The revised edition of this book offers an extended overview of quantum walks and explains their role in building quantum algorithms, in particular search algorithms.

Updated throughout, the book focuses on core topics including Grover's algorithm and the most important quantum walk models, such as the coined, continuous-time, and Szedgedy's quantum walk models. There is a new chapter describing the staggered quantum walk model. The chapter on spatial search algorithms has been rewritten to offer a more comprehensive approach and a new chapter describing the element distinctness algorithm has been added. There is a new appendix on graph theory highlighting the importance of graph theory to quantum walks.

As before, the reader will benefit from the pedagogical elements of the book, which include exercises and references to deepen the reader's understanding, and guidelines for the use of computer programs to simulate the evolution of quantum walks.

Review of the first edition:

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.

Offers an expanded introduction to the field of quantum walks Covers key topics in quantum computation including Grover's algorithm Includes a new chapter on the Element Distinctness Algorithm

Autorentext

Renato Portugal graduated in Physics from the Pontifical Catholic University of Rio de janeiro in 1981 and obtained his PhD degree in Physics at the Brazilian Center for Research in Physics in 1988. He was a visiting professor at the University of Waterloo in 1997 and 2008 and at the Queen's University at Kingston in 1998 in Canada. He is currently a full researcher at the National Laboratory of Scientific Computing (LNCC). Currently, he is working in the area of quantum computing with focus on the following subareas: algorithms for quantum computing, analysis and simulation of quantum walks, and classical cryptography. Franklin de Lima Marquezino graduated in Computer Science from the Catholic University of Petropolis in 2004, and received his PhD degree in Computer Modelling form the National Laboratory of Scientific Computing (LNCC) in 2010. He stayed for one year as postdoctoral researcher also at LNCC. Since 2011, he is an associate professor at the Federal University of Rio de Janeiro, working mainly in the areas of quantum algorithms and quantum walks. Carlile Lavor graduated in Mathematics from the University of Campinas in 1996, and received a PhD in Computer Science from the Federal University of Rio de Janeiro, in 2001. He was a visiting professor at prestigious institutions like École Polytechnique (2008-2009) and Duke University (2013-2014). Since 2005, he has been working at the University of Campinas, where he is now a full professor. He is co-author of the books "Euclidean Distance Geometry" and "A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry" and co-editor of "Distance Geometry: Theory, Methods and Applications" book, all by Springer.

Inhalt

1 Introduction.- 2 The Postulates of Quantum Mechanics.- 3 Introduction to Quantum Walks.- 4 Grover's Algorithm and Its Generalization.- 5 Coined Walks on Infinite Lattices.- 6 Coined Walks with Cyclic Boundary Conditions.- 7 Coined Quantum Walks on Graphs.- 8 Staggered Model.- 9 Spatial Search Algorithms.- 10 Element Distinctness.- 11 Szegedy's Quantum Walk.- A Linear Algebra for Quantum Computation.- B Graph Theory for Quantum Walk.- C Classical Hitting Time.