Keywords:
Quantum theory-Data processing.
;
Electronic books.
Type of Medium:
Online Resource
Pages:
1 online resource (314 pages)
Edition:
2nd ed.
ISBN:
9783319978130
Series Statement:
Quantum Science and Technology Series
URL:
https://ebookcentral.proquest.com/lib/geomar/detail.action?docID=5495996
DDC:
530.1201/51
Language:
English
Note:
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 Walks -- 3.5.1 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 Lattices -- 5.1 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 Segment -- 7 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 Lattices -- 7.6 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 -- 9.5 Grover's Algorithm as Spatial Search on Graphs -- 9.5.1 Grover's Algorithm in terms of the Coined Model -- 9.5.2 Grover's Algorithm in terms of the Staggered Model -- 9.5.3 Complexity Analysis of Grover's Algorithm -- 10 Element Distinctness -- 10.1 Classical Algorithms -- 10.2 Naïve Quantum Algorithms -- 10.3 The Optimal Quantum Algorithm -- 10.3.1 Analysis of the Algorithm -- 10.3.2 Number of Queries -- 10.3.3 Example -- 11 Szegedy's Quantum Walk -- 11.1 Discrete-Time Markov Chains -- 11.2 Markov Chain-Based Quantum Walk -- 11.3 Evolution Operator -- 11.4 Singular Values and Vectors of the Discriminant -- 11.5 Eigenvalues and Eigenvectors of the Evolution Operator.
,
11.6 Quantum Hitting Time -- 11.7 Searching Instead of Detecting -- 11.8 Example: Complete Graphs -- 11.8.1 Probability of Finding a Marked Element -- A Linear Algebra for Quantum Computation -- A.1 Vector Spaces -- A.2 Inner Product -- A.3 The Dirac Notation -- A.4 Computational Basis -- A.5 Qubit and the Bloch Sphere -- A.6 Linear Operators -- A.7 Matrix Representation -- A.8 Diagonal Representation -- A.9 Completeness Relation -- A.10 Cauchy-Schwarz Inequality -- A.11 Special Operators -- A.12 Pauli Matrices -- A.13 Operator Functions -- A.14 Norm of a Linear Operator -- A.15 Tensor Product -- A.16 Quantum Gates, Circuits, and Registers -- Appendix B Graph Theory for Quantum Walks -- B.1 Basic Definitions -- B.2 Multigraph -- B.3 Bipartite Graph -- B.4 Intersection Graph -- B.5 Clique, Stable Set, and Matching -- B.6 Graph Operators -- B.6.1 Clique Graph Operator -- B.6.2 Line Graph Operator -- B.6.3 Subdivision Graph Operator -- B.6.4 Clique-Insertion Operator -- B.7 Coloring -- B.8 Diameter -- B.9 Directed Graph -- B.10 Some Named Graphs -- B.10.1 Johnson Graphs -- B.10.2 Kneser Graphs -- B.10.3 Cayley Graphs -- Appendix C Classical Hitting Time -- C.1 Hitting Time Using the Stationary Distribution -- C.2 Hitting Time Without the Stationary Distribution -- Appendix References -- -- Index.
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