This book presents various theories and algorithms to create a quantum computer. The concept of the classical and quantum computers, and the concept of circuits and gates are reviewed. The example of the Deutsch and the Deutsch-Josca algorithm is discussed to illustrate some key features of quantum computing. The Grover algorithm, considered to be of major milestone of the subject, is discussed in detail to exemplify the techniques used in computer algorithms. The role of quantum superposition (also called quantum parallelism) and of quantum entanglement is discussed in order to understand the key advantages of a quantum over a classical computer.
Author(s): Belal Ehsan Baaquie, Leong-Chuan Kwek
Publisher: Springer
Year: 2023
Language: English
Pages: 296
City: Singapore
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
