Quantum computing : a gentle introduction /
A thorough exposition of quantum computing and the underlying concepts of quantum physics, with explanations of the relevant mathematics and numerous examples.
Clasificación: | Libro Electrónico |
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Autor principal: | |
Otros Autores: | |
Formato: | Electrónico eBook |
Idioma: | Inglés |
Publicado: |
Cambridge, Mass. :
MIT Press,
2011.
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Colección: | Scientific and engineering computation.
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Temas: | |
Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.