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Quantum Computing Explained

Book Description

A self-contained treatment of the fundamentals of quantum computing

This clear, practical book takes quantum computing out of the realm of theoretical physics and teaches the fundamentals of the field to students and professionals who have not had training in quantum computing or quantum information theory, including computer scientists, programmers, electrical engineers, mathematicians, physics students, and chemists. The author cuts through the conventions of typical jargon-laden physics books and instead presents the material through his unique "how-to" approach and friendly, conversational style.

Readers will learn how to carry out calculations with explicit details and will gain a fundamental grasp of:

*Quantum mechanics

*Quantum computation

*Teleportation

*Quantum cryptography

*Entanglement

*Quantum algorithms

*Error correction

A number of worked examples are included so readers can see how quantum computing is done with their own eyes, while answers to similar end-of-chapter problems are provided for readers to check their own work as they learn to master the information.

Ideal for professionals and graduate-level students alike, Quantum Computing Explained delivers the fundamentals of quantum computing readers need to be able to understand current research papers and go on to study more advanced quantum texts.

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