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Quantum walks and search algorithms

Author: Renato Portugal
Publisher: Cham : Springer, [2018] ©2018
Series: Quantum science and technology.
Edition/Format:   eBook : Document : English : Second editionView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Portugal, Renato.
Quantum walks and search algorithms.
Cham : Springer, [2018]
(OCoLC)1042354468
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Renato Portugal
ISBN: 9783319978130 3319978136 9783319978147 3319978144
OCLC Number: 1049974756
Description: 1 online resource : illustrations
Contents: Intro; Preface; Acknowledgements; Contents; 1 Introduction; 2 The Postulates of Quantum Mechanics; 2.1 State Space; 2.1.1 State Space Postulate; 2.2 Unitary Evolution; 2.2.1 Evolution Postulate; 2.3 Composite Systems; 2.4 Measurement Process; 2.4.1 Measurement Postulate; 2.4.2 Measurement in the Computational Basis; 2.4.3 Partial Measurement in the Computational Basis; 3 Introduction to Quantum Walks; 3.1 Classical Random Walk on the Line; 3.2 Classical Discrete-Time Markov Chains; 3.3 Coined Quantum Walks; 3.3.1 Coined Walk on the Line; 3.4 Classical Continuous-Time Markov Chains. 3.5 Continuous-Time Quantum Walks3.5.1 Continuous-Time Walk on the Line; 3.5.2 Why Must Time be Continuous?; 4 Grover's Algorithm and Its Generalization; 4.1 Grover's Algorithm; 4.2 Quantum Circuit of Grover's Algorithm; 4.3 Analysis of the Algorithm Using Reflection Operators; 4.4 Analysis Using the Two-Dimensional Real Space; 4.5 Analysis Using the Spectral Decomposition; 4.6 Optimality of Grover's Algorithm; 4.7 Search with Repeated Elements; 4.7.1 Analysis Using Reflection Operators; 4.7.2 Analysis Using the Reduced Space; 4.8 Amplitude Amplification; 4.8.1 The Technique. 5 Coined Walks on Infinite Lattices5.1 Hadamard Walk on the Line; 5.1.1 Fourier Transform; 5.1.2 Analytic Solution; 5.1.3 Other Coins; 5.2 Two-Dimensional Lattice; 5.2.1 The Hadamard Coin; 5.2.2 The Fourier Coin; 5.2.3 The Grover Coin; 5.2.4 Standard Deviation; 5.3 Quantum Walk Packages; 6 Coined Walks with Cyclic Boundary Conditions; 6.1 Cycles; 6.1.1 Fourier Transform; 6.1.2 Analytic Solutions; 6.1.3 Periodic Solutions; 6.2 Finite Two-Dimensional Lattices; 6.2.1 Fourier Transform; 6.2.2 Analytic Solutions; 6.3 Hypercubes; 6.3.1 Fourier Transform; 6.3.2 Analytic Solutions. 6.3.3 Reducing a Hypercube to a Line Segment7 Coined Quantum Walks on Graphs; 7.1 Quantum Walks on Class-1 Regular Graphs; 7.2 Coined Quantum Walks on Arbitrary Graphs; 7.2.1 Locality; 7.2.2 Grover Quantum Walk; 7.2.3 Coined Walks on Cayley Graphs; 7.2.4 Coined Walks on Multigraphs; 7.3 Dynamics and Quasi-periodicity; 7.4 Perfect State Transfer and Fractional Revival; 7.5 Limiting Probability Distribution; 7.5.1 Limiting Distribution Using the Fourier Basis; 7.5.2 Limiting Distribution of QWs on Cycles; 7.5.3 Limiting Distribution of QWs on Hypercubes. 7.5.4 Limiting Distribution of QWs on Finite Lattices7.6 Distance Between Distributions; 7.7 Mixing Time; 7.7.1 Instantaneous Uniform Mixing (IUM); 8 Staggered Model; 8.1 Graph Tessellation Cover; 8.2 The Evolution Operator; 8.3 Staggered Walk on the Line; 8.3.1 Fourier Analysis; 8.3.2 Standard Deviation; 9 Spatial Search Algorithms; 9.1 Quantum-Walk-Based Search Algorithms; 9.2 Analysis of the Time Complexity; 9.2.1 Case B=0; 9.2.2 Tulsi's Modification; 9.3 Finite Two-Dimensional Lattices; 9.3.1 Tulsi's Modification of the Two-Dimensional Lattice; 9.4 Hypercubes.
Series Title: Quantum science and technology.
Responsibility: Renato Portugal.

Abstract:

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  Read more...

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