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Table of 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.
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.