Quantum Computers: Theory and Algorithms

Preface
	References
Acknowledgments
Contents
About the Authors
1 Introduction
	1.1 Interview on Quantum Computers
	References
Part I Fundamentals
2 Binary Numbers, Vectors, Matrices and Tensor Products
	2.1 Binary Representation
	2.2 Linear Vector Space
	2.3 N-Dimensional Complex Linear Vector Space
	2.4 Matrices
	2.5 Properties of N timesN Matrices
		2.5.1 Hermitian Conjugation
	2.6 Tensor (Outer) Product
	2.7 Square Matrices
	2.8 Dirac Bracket: Vector Notation
	2.9 Tensor and Outer Product: Strings and Gates
		2.9.1 3-Bits String
	References
3 Classical Gates and Algorithms
	3.1 Classical Algorithm
	3.2 Classical Gates
	3.3 2-Bits String Gates
	3.4 XOR Reversible Gate
		3.4.1 XOR: 3-Bits String
	3.5 3-Bits String: Toffoli Gate
	3.6 Unitary AND, NAND and NOT Gates for 3-Bits
	3.7 Unitary OR and NOR Gates
	3.8 Classical Binary Addition
	3.9 Half-Adder
	3.10 Full-Adder
	3.11 Matrices of Full-Adder Gates
	References
4 Principles of Quantum Mechanics
	4.1 Degrees of Freedom: Indeterminate
	4.2 Hilbert Space; State Vectors
		4.2.1 Continuous Degrees of Freedom
		4.2.2 Discrete Degrees of Freedom: Qubits
	4.3 Hermitian and Unitary Operators
	4.4 The Schrödinger Equation
		4.4.1 Key Features of the Schrödinger Equation
	4.5 Quantum Measurement: Born Rule
	4.6 Quantum Measurements and Degrees of Freedom
	4.7 No-Cloning Theorem
	4.8 Copenhagen Interpretation: Open Questions
	4.9 Summary of Quantum Mechanics
	4.10 Generalized Born Rule
		4.10.1 Example
	4.11 Consistency of Generalized Born Rule
	4.12 Quantum Mechanics and Quantum Computers
	References
5 Quantum Superposition and Entanglement
	5.1 Quantum Superposition
		5.1.1 The Experiment
		5.1.2 Experiment with Detectors 1 and 2: No Interference
		5.1.3 Experiment Without Detectors 1 and 2: Indeterminate
	5.2 Quantum Superposition and Quantum Algorithms
	5.3 Partial Trace for Tensor Products
	5.4 Density Matrix ρ
		5.4.1 Pure Density Matrix
		5.4.2 Mixed Density Matrix
		5.4.3 Density Matrix for a Two-State System
	5.5 Reduced Density Matrix
	5.6 Separable Quantum Systems
	5.7 Entangled Quantum States
	5.8 Entanglement for Composite Systems
	5.9 Entangled State: Two Binary Degrees of Freedom
	5.10 Quantum Entropy
	5.11 Maximally Entangled States
		5.11.1 An Entangled State of Two Binary Degrees of Freedom
	5.12 Pure and Mixed Density Matrix
	References
6 Binary Degrees of Freedom and Qubits
	6.1 Introduction
	6.2 Degrees of Freedom and Qubits
	6.3 Single Qubit
		6.3.1 Density Matrix
	6.4 Bell Entangled Qubits
	6.5 Bell States: Maximally Entangled
7 Quantum Gates and Circuits
	7.1 Quantum Gates
	7.2 Superposed and Entangled Qubits
	7.3 Two- and Three-Qubit Quantum Gates
	7.4 Arithmetic Addition of Binary Qubits
	7.5 Quantum Measurements of Qubits
		7.5.1 Partial Measurement
	References
8 Phase Estimation and quantum Fourier Transform (qFT)
	8.1 Introduction
	8.2 Eigenvalue of Unitary Operator
	8.3 Phase Estimation
		8.3.1 Phase Estimation for n=3
	8.4 quantum Fourier Transform
		8.4.1 quantum Fourier Transform for n=3
	8.5 Quantum Circuit of qFT
	References
Part II Quantum Algorithms
9 Deutsch Algorithm
	9.1 The Deutsch Quantum Circuit
10 Deutsch–Jozsa Algorithm
11 Grover's Algorithm
	11.1 Phase Inversion and Amplitude Amplification
	11.2 Grover's Quantum Circuit
	11.3 Grover Algorithm: Two-Qubit
	11.4 Grover Algorithm: Phase Inversion
	11.5 Grover Diffusion Gate W
	11.6 Grover Recursion Equation
	11.7 Single Recursion: Two Qubits
	11.8 Discussion
	References
12 Simon's Algorithm
	12.1 Quantum Algorithm
	12.2 An Illustrative Example
	Reference
13 Shor's Algorithm
	13.1 Introduction
	13.2 Understanding the Classical Algorithm
	13.3 Quantum Algorithm
		13.3.1 How Then Does the Quantum Algorithm Work?
	References
Part III Applications
14 Quantum Algorithm for Option Pricing
	14.1 Review of Option Pricing
	14.2 Quantum Algorithm
	14.3 Quantum Algorithm for Expectation Value
	14.4 Algorithm for Quadratic Improvement
	14.5 Eigenvalues of Diffusion Operator Q
	14.6 Amplitude Amplification
	14.7 Call Option
	14.8 Discussion
	References
15 Solving Linear Equations
	15.1 Introduction
	15.2 Harrow–Hassidim–Lloyd Algorithm
	15.3 Specific Example
	15.4 Why Do We Not Need the Eigenvalues?
	15.5 Other Applications
	References
16 Quantum-Classical Hybrid Algorithms
	16.1 Why Bother?
	16.2 Variational Quantum Eigensolvers
	16.3 Quantum Approximate Optimization Algorithm
	16.4 MaxCut Problem
	References
17 Quantum Error Correction
	17.1 Introduction
	17.2 Simple Quantum Errors
	17.3 Kraus Operators
	17.4 Nine-qubit Code
	17.5 General Properties of Quantum Error-Correcting Codes
	17.6 Classical Linear Codes
	17.7 An Example of a Linear Code: Hamming Code
	17.8 Quantum Linear Codes: CSS Codes
	References
18 One-Way Quantum Computer
	18.1 Measurement-Based Quantum Computation
	18.2 The Cluster State
	18.3 Simulation of Basic Quantum Gates
	18.4 Resource States for MBQC
	References
Part IV Summary
19 Efficiency of a Quantum Computer
	19.1 Quantum Algorithms
	19.2 Memory and Speed of Quantum Computations
	19.3 Where Does Quantum Computation Take Place?
	19.4 Conclusions
	References
Index