Quantum Computing Explained

Table of Contents

1. Cover
2. Halftitle
3. Title page
4. Copyright
5. Contents
6. Preface
7. 1 A BRIEF INTRODUCTION TO INFORMATION THEORY
   1. Classical Information
   2. Information Content in a Signal
   3. Entropy and Shannon’S Information Theory
   4. Probability Basics
   5. EXERCISES
8. 2 QUBITS AND QUANTUM STATES
   1. The Qubit
   2. Vector Spaces
   3. Linear Combinations of Vectors
   4. Uniqueness of a Spanning Set
   5. Basis and Dimension
   6. Inner Products
   7. Orthonormality
   8. Gram-Schmidt Orthogonalization
   9. Bra-Ket Formalism
   10. The Cauchy-Schwartz And Triangle Inequalities
   11. Summary
   12. Exercises
9. 3 MATRICES AND OPERATORS
   1. Observables
   2. The Pauli Operators
   3. Outer Products
   4. The Closure Relation
   5. Representations of Operators Using Matrices
   6. Outer Products and Matrix Representations
   7. Matrix Representation of Operators in Two-Dimensional Spaces
   8. Definition: The Pauli Matrices
   9. Hermitian, Unitary, And Normal Operators
   10. Eigenvalues and Eigenvectors
   11. Spectral Decomposition
   12. The Trace Of An Operator
   13. Important Properties of the Trace
   14. The Expectation Value of an Operator
   15. Projection Operators
   16. POSITIVE OPERATORS
   17. Commutator Algebra
   18. The Heisenberg Uncertainty Principle
   19. Polar Decomposition and Singular Values
   20. The Postulates of Quantum Mechanics
   21. Exercises
10. 4 TENSOR PRODUCTS
    1. Representing Composite States in Quantum Mechanics
    2. Computing Inner Products
    3. Tensor Products of Column Vectors
    4. Operators and Tensor Products
    5. Tensor Products of Matrices
    6. Exercises
11. 5 THE DENSITY OPERATOR
    1. The Density Operator For a Pure State
    2. The Density Operator For a Mixed State
    3. Key Properties of a Density Operator
    4. Characterizing Mixed States
    5. The Partial Trace And The Reduced Density Operator
    6. The Density Operator and the Bloch Vector
    7. EXERCISES
12. 6 QUANTUM MEASUREMENT THEORY
    1. Distinguishing Quantum States and Measurement
    2. Projective Measurements
    3. Measurements on Composite Systems
    4. Generalized Measurements
    5. Positive Operator-Valued Measures
    6. Exercises
13. 7 ENTANGLEMENT
    1. Bell’S Theorem
    2. Bipartite Systems and the Bell Basis
    3. When is a State Entangled?
    4. The Pauli Representation
    5. Entanglement Fidelity
    6. Using Bell States For Density Operator Representation
    7. Schmidt Decomposition
    8. Purification
    9. Exercises
14. 8 QUANTUM GATES AND CIRCUITS
    1. Classical Logic Gates
    2. Single-Qubit Gates
    3. More Single-Qubit Gates
    4. Exponentiation
    5. The Z–Y Decomposition
    6. Basic Quantum Circuit Diagrams
    7. Controlled Gates
    8. Gate Decomposition
    9. Exercises
15. 9 QUANTUM ALGORITHMS
    1. Hadamard Gates
    2. The Phase Gate
    3. Matrix Representation of Serial and Parallel Operations
    4. Quantum Interference
    5. Quantum Parallelism and Function Evaluation
    6. Deutsch-Jozsa Algorithm
    7. Quantum Fourier Transform
    8. Phase Estimation
    9. Shor’S Algorithm
    10. Quantum Searching And Grover’S Algorithm
    11. Exercises
16. 10 APPLICATIONS OF ENTANGLEMENT: TELEPORTATION AND SUPERDENSE CODING
    1. Teleportation
    2. The Peres Partial Transposition Condition
    3. Entanglement Swapping
    4. Superdense Coding
    5. Exercises
17. 11 QUANTUM CRYPTOGRAPHY
    1. A Brief Overview of Rsa Encryption
    2. Basic Quantum Cryptography
    3. An Example Attack: The Controlled Not Attack
    4. The B92 Protocol
    5. The E91 Protocol (Ekert)
    6. Exercises
18. 12 QUANTUM NOISE AND ERROR CORRECTION
    1. Single-Qubit Errors
    2. Quantum Operations And Krauss Operators
    3. The Depolarization Channel
    4. The Bit Flip and Phase Flip Channels
    5. Amplitude Damping
    6. Phase Damping
    7. Quantum Error Correction
    8. Exercises
19. 13 TOOLS OF QUANTUM INFORMATION THEORY
    1. The No-Cloning Theorem
    2. Trace Distance
    3. Fidelity
    4. Entanglement of Formation and Concurrence
    5. Information Content and Entropy
    6. Exercises
20. 14 ADIABATIC QUANTUM COMPUTATION
    1. Adiabatic Processes
    2. Adiabatic Quantum Computing
    3. Exercises
21. 15 CLUSTER STATE QUANTUM COMPUTING
    1. Cluster States
    2. Adjacency Matrices
    3. Stabilizer States
    4. Aside: Entanglement Witness
    5. Cluster State Processing
    6. Exercises
22. References
23. Index