Quantum computing : a gentle introduction / Eleanor Rieffel and Wolfgang Polak.

A thorough exposition of quantum computing and the underlying concepts of quantum physics, with explanations of the relevant mathematics and numerous examples.

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Bibliographic Details
Main Author:	Rieffel, Eleanor, 1965-
Other Authors:	Polak, Wolfgang, 1950-
Format:	eBook
Language:	English
Published:	Cambridge, Mass. : MIT Press, 2011.
Series:	Scientific and engineering computation.
Subjects:	Quantum computers. Quantum theory. COMPUTERS > Hardware > Mainframes & Minicomputers. Quantum computers Quantum theory Kvantdatorer. Kvantteori.
Online Access:	Click for online access

Table of Contents:

Machine generated contents note: 1. Introduction
I. QUANTUM BUILDING BLOCKS
2. Single-Qubit Quantum Systems
2.1. The Quantum Mechanics of Photon Polarization
2.1.1. A Simple Experiment
2.1.2. A Quantum Explanation
2.2. Single Quantum Bits
2.3. Single-Qubit Measurement
2.4. A Quantum Key Distribution Protocol
2.5. The State Space of a Single-Qubit System
2.5.1. Relative Phases versus Global Phases
2.5.2. Geometric Views of the State Space of a Single Qubit
2.5.3. Comments on General Quantum State Spaces
2.6. References
2.7. Exercises
3. Multiple-Qubit Systems
3.1. Quantum State Spaces
3.1.1. Direct Sums of Vector Spaces
3.1.2. Tensor Products of Vector Spaces
3.1.3. The State Space of an n-Qubit System
3.2. Entangled States
3.3. Basics of Multi-Qubit Measurement
3.4. Quantum Key Distribution Using Entangled States
3.5. References
3.6. Exercises
4. Measurement of Multiple-Qubit States
4.1. Dirac's Bra/Ket Notation for Linear Transformations.
4.2. Projection Operators for Measurement
4.3. Hermitian Operator Formalism for Measurement
4.3.1. The Measurement Postulate
4.4. EPR Paradox and Bell's Theorem
4.4.1. Setup for Bell's Theorem
4.4.2. What Quantum Mechanics Predicts
4.4.3. Special Case of Bell's Theorem: What Any Local Hidden Variable Theory Predicts
4.4.4. Bell's Inequality
4.5. References
4.6. Exercises
5. Quantum State Transformations
5.1. Unitary Transformations
5.1.1. Impossible Transformations: The No-Cloning Principle
5.2. Some Simple Quantum Gates
5.2.1. The Pauli Transformations
5.2.2. The Hadamard Transformation
5.2.3. Multiple-Qubit Transformations from Single-Qubit Transformations
5.2.4. The Controlled-not and Other Singly Controlled Gates
5.3. Applications of Simple Gates
5.3.1. Dense Coding
5.3.2. Quantum Teleportation
5.4. Realizing Unitary Transformations as Quantum Circuits
5.4.1. Decomposition of Single-Qubit Transformations
5.4.2. Singly-Controlled Single-Qubit Transformations
5.4.3. Multiply-Controlled Single-Qubit Transformations.
5.4.4. General Unitary Transformations
5.5. A Universally Approximating Set of Gates
5.6. The Standard Circuit Model
5.7. References
5.8. Exercises
6. Quantum Versions of Classical Computations
6.1. From Reversible Classical Computations to Quantum Computations
6.1.1. Reversible and Quantum Versions of Simple Classical Gates
6.2. Reversible Implementations of Classical Circuits
6.2.1. A Naive Reversible Implementation
6.2.2. A General Construction
6.3. A Language for Quantum Implementations
6.3.1. The Basics
6.3.2. Functions
6.4. Some Example Programs for Arithmetic Operations
6.4.1. Efficient Implementation of and
6.4.2. Efficient Implementation of Multiply-Controlled Single-Qubit Transformations
6.4.3. In-Place Addition
6.4.4. Modular Addition
6.4.5. Modular Multiplication
6.4.6. Modular Exponentiation
6.5. References
6.6. Exercises
II. QUANTUM ALGORITHMS
7. Introduction to Quantum Algorithms
7.1. Computing with Superpositions
7.1.1. The Walsh-Hadamard Transformation.
7.1.2. Quantum Parallelism
7.2. Notions of Complexity
7.2.1. Query Complexity
7.2.2. Communication Complexity
7.3. A Simple Quantum Algorithm
7.3.1. Deutsch's Problem
7.4. Quantum Subroutines
7.4.1. The Importance of Unentangling Temporary Qubits in Quantum Subroutines
7.4.2. Phase Change for a Subset of Basis Vectors
7.4.3. State-Dependent Phase Shifts
7.4.4. State-Dependent Single-Qubit Amplitude Shifts
7.5. A Few Simple Quantum Algorithms
7.5.1. Deutsch-Jozsa Problem
7.5.2. Bernstein-Vazirani Problem
7.5.3. Simon's Problem
7.5.4. Distributed Computation
7.6. Comments on Quantum Parallelism
7.7. Machine Models and Complexity Classes
7.7.1. Complexity Classes
7.7.2. Complexity: Known Results
7.8. Quantum Fourier Transformations
7.8.1. The Classical Fourier Transform
7.8.2. The Quantum Fourier Transform
7.8.3. A Quantum Circuit for Fast Fourier Transform
7.9. References
7.10. Exercises
8. Shor's Algorithm
8.1. Classical Reduction to Period-Finding.
8.2. Shor's Factoring Algorithm
8.2.1. The Quantum Core
8.2.2. Classical Extraction of the Period from the Measured Value
8.3. Example Illustrating Shor's Algorithm
8.4. The Efficiency of Shor's Algorithm
8.5. Omitting the Internal Measurement
8.6. Generalizations
8.6.1. The Discrete Logarithm Problem
8.6.2. Hidden Subgroup Problems
8.7. References
8.8. Exercises
9. Graver's Algorithm and Generalizations
9.1. Graver's Algorithm
9.1.1. Outline
9.1.2. Setup
9.1.3. The Iteration Step
9.1.4. How Many Iterations?
9.2. Amplitude Amplification
9.2.1. The Geometry of Amplitude Amplification
9.3. Optimality of Grover's Algorithm
9.3.1. Reduction to Three Inequalities
9.3.2. Proofs of the Three Inequalities
9.4. Derandomization of Grover's Algorithm and Amplitude Amplification
9.4.1. Approach 1: Modifying Each Step
9.4.2. Approach 2: Modifying Only the Last Step
9.5. Unknown Number of Solutions
9.5.1. Varying the Number of Iterations
9.5.2. Quantum Counting.
9.6. Practical Implications of Grover's Algorithm and Amplitude Amplification
9.7. References
9.8. Exercises
III. ENTANGLED SUBSYSTEMS AND ROBUST QUANTUM COMPUTATION
10. Quantum Subsystems and Properties of Entangled States
10.1. Quantum Subsystems and Mixed States
10.1.1. Density Operators
10.1.2. Properties of Density Operators
10.1.3. The Geometry of Single-Qubit Mixed States
10.1.4. Von Neumann Entropy
10.2. Classifying Entangled States
10.2.1. Bipartite Quantum Systems
10.2.2. Classifying Bipartite Pure States up to LOCC Equivalence
10.2.3. Quantifying Entanglement in Bipartite Mixed States
10.2.4. Multipartite Entanglement
10.3. Density Operator Formalism for Measurement
10.3.1. Measurement of Density Operators
10.4. Transformations of Quantum Subsystems and Decoherence
10.4.1. Superoperators
10.4.2. Operator Sum Decomposition
10.4.3. A Relation Between Quantum State Transformations and Measurements
10.4.4. Decoherence
10.5. References
10.6. Exercises
11. Quantum Error Correction.
11.1. Three Simple Examples of Quantum Error Correcting Codes
11.1.1. A Quantum Code That Corrects Single Bit-Flip Errors
11.1.2. A Code for Single-Qubit Phase-Flip Errors
11.1.3. A Code for All Single-Qubit Errors
11.2. Framework for Quantum Error Correcting Codes
11.2.1. Classical Error Correcting Codes
11.2.2. Quantum Error Correcting Codes
11.2.3. Correctable Sets of Errors for Classical Codes
11.2.4. Correctable Sets of Errors for Quantum Codes
11.2.5. Correcting Errors Using Classical Codes
11.2.6. Diagnosing and Correcting Errors Using Quantum Codes
11.2.7. Quantum Error Correction across Multiple Blocks
11.2.8. Computing on Encoded Quantum States
11.2.9. Superpositions and Mixtures of Correctable Errors Are Correctable
11.2.10. The Classical Independent Error Model
11.2.11. Quantum Independent Error Models
11.3. CSS Codes
11.3.1. Dual Classical Codes
11.3.2. Construction of CSS Codes from Classical Codes Satisfying a Duality Condition
11.3.3. The Steane Code
11.4. Stabilizer Codes.
13.4. Alternatives to the Circuit Model of Quantum Computation
13.4.1. Measurement-Based Cluster State Quantum Computation
13.4.2. Adiabatic Quantum Computation
13.4.3. Holonomic Quantum Computation
13.4.4. Topological Quantum Computation
13.5. Quantum Protocols
13.6. Insight into Classical Computation
13.7. Building Quantum Computers
13.8. Simulating Quantum Systems
13.9. Where Does the Power of Quantum Computation Come From?
13.10. What if Quantum Mechanics Is Not Quite Correct?
APPENDIXES
A. Some Relations Between Quantum Mechanics and Probability Theory
A.1. Tensor Products in Probability Theory
A.2. Quantum Mechanics as a Generalization of Probability Theory
A.3. References
A.4. Exercises
B. Solving the Abelian Hidden Subgroup Problem.
B.1. Representations of Finite Abelian Groups
B.1.1. Schur's Lemma
B.2. Quantum Fourier Transforms for Finite Abelian Groups
B.2.1. The Fourier Basis of an Abelian Group
B.2.2. The Quantum Fourier Transform Over a Finite Abelian Group
B.3. General Solution to the Finite Abelian Hidden Subgroup Problem
B.4. Instances of the Abelian Hidden Subgroup Problem
B.4.1. Simon's Problem
B.4.2. Shor's Algorithm: Finding the Period of a Function
B.5. Comments on the Non-Abelian Hidden Subgroup Problem
B.6. References
B.7. Exercises.

Quantum computing : a gentle introduction / Eleanor Rieffel and Wolfgang Polak.

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