Table of Contents
Acknowledgments xi
Introduction xiii
1 Spin 1
The Quantum Clock 6
Measurements in the Same Direction 7
Measurements in Different Directions 7
Measurements 9
Randomness 10
Photons and Polarization 11
Conclusions 15
2 Linear Algebra 17
Complex Numbers versus Real Numbers 17
Vectors 19
Diagrams of Vectors 19
Lengths of Vectors 20
Scalar Multiplication 21
Vector Addition 21
Orthogonal Vectors 23
Multiplying a Bra by a Ket 23
Bra-Kets and Lengths 24
Bra-Kets and Orthogonality 24
Orthonormal Bases 25
Vectors as Linear Combinations of Basts Vectors 27
Ordered Bases 29
Length of Vectors 30
Matrices 30
Matrix Computations 33
Orthogonal and Unitary Matrices 34
Linear Algebra Toolbox 35
3 Spin and Qubits 37
Probability 37
Mathematics of Quantum Spin 38
Equivalent State Vectors 41
The Basis Associated with a Given Spin Direction 43
Rotating the Apparatus through 60° 45
The Mathematical Model for Photon Polarization 46
The Basis Associated with a Given Polarization Direction 47
The Polarized Filters Experiments 47
Qubits 49
Alice, Bob, and Eve 50
Probability Amplitudes and Interference 52
Alice, Bob, Eve, and the BB84 Protocol 53
4 Entanglement 57
Alice and Bob's Qubits Are Not Entangled 57
Unentangled Qubits Calculation 59
Entangled Qubits Calculation 61
Superluminal Communication 62
The Standard Basis for Tensor Products 64
How Do You Entangle Qubits? 65
Using the CNOT Gate to Entangle Qubits 67
Entangled Quantum Clocks 68
5 Bell's Inequality 71
Entangled Qubits in Different Bases 72
Proof That Equals 73
Einstein and Local Realism 75
Einstein and Hidden Variables 77
A Classical Explanation of Entanglement 78
Bell's Inequality 79
The Answer of Quantum Mechanics 80
The Classical Answer 81
Measurement 84
The Ekert Protocol for Quantum Key Distribution 86
6 Classical Logic, Gates, and Circuits 89
Logic 90
Boolean Algebra 91
Functional Completeness 94
Gates 98
Circuits 99
NAND Is a Universal Gate 100
Gates and Computation 101
Memory 103
Reversible Computation 103
Billiard Ball Computing 111
7 Quantum Gates and Circuits 117
Qubits 118
The CNOT Gate 118
Quantum Gates 120
Quantum Gates Acting on One Qubit 121
Are There Universal Quantum Gates? 123
No Cloning Theorem 124
Quantum Computation versus Classical Computation 126
The Bell Circuit 127
Superdense Coding 129
Quantum Teleportation 132
Error Correction 135
8 Quantum Algorithms 141
The Complexity Classes P and NP 142
Are Quantum Algorithms Faster Than Classical Ones? 144
Query Complexity 145
Deutsch's Algorithm 145
The Kronecker Product of Hadamard Matrices 149
The Deutsch-Jozsa Algorithm 152
Simon's Algorithm 157
Complexity Classes 166
Quantum Algorithms 168
9 Impact of Quantum Computing 171
Shor's Algorithm and Cryptanalysis 172
Grover's Algorithm and Searching Data 176
Chemistry and Simulation 181
Hardware 182
Quantum Supremacy and Parallel Universes 186
Computation 187
Index 191