A First Introduction to Quantum Computing and Information

Foreword
Preface
	On Notation
Acknowledgements
Contents
1 A Quantum Mechanic's Toolbox
	1.1 Bits and Qubits
		1.1.1 Binary Arithmetic
	1.2 A Short Introduction to Linear Vector Spaces
	1.3 Hilbert Space
		1.3.1 Dirac's Bra-Ket Notation
		1.3.2 Outer Products and Operators
		1.3.3 Direct and Kronecker Products
	Problems
2 Apples and Oranges: Matrix Representations
	2.1 Matrix Representations
		2.1.1 Matrix Operations
		2.1.2 The Bloch sphere
	2.2 The Pauli Matrices
	2.3 Polarization of Light: A Classical Qubit
		2.3.1 A Qubit Parable
	2.4 Spin
		2.4.1 Non-commuting Observables and the Uncertainty Principle
	2.5 Direct Products
	Problems
3 Circuit Model of Computation
	3.1 Boole's Logic Tables
		3.1.1 Gates as Mappings
	3.2 Our First Quantum Circuit
		3.2.1 Multi-qubit Gates
		3.2.2 Deutsch's Algorithm
		3.2.3 Deutsch-Josza Algorithm
	3.3 Hamiltonian Evolution
	Problems
4 Quantum Killer Apps: Quantum Fourier Transform and Search Algorithms
	4.1 Introduction
	4.2 Fourier Series
		4.2.1 Nyquist-Shannon Sampling
		4.2.2 Discrete Fourier Transform
	4.3 Quantum Fourier Transform
		4.3.1 QFT Diagrammatics
		4.3.2 Period Finding with the bold upper Q bold upper F bold upper T QFT  Gate
		4.3.3 Shor's Algorithm
	4.4 Grover's Search Algorithm
	Problems
5 Quantum Mechanics According to Martians: Density Matrix Theory
	5.1 Introduction
	5.2 Density Operators and Matrices
	5.3 Pure and Mixed States
	5.4 Reduced Density Operators
		5.4.1 Entangled States
	5.5 Schmidt Decomposition
	5.6 von Neumann Entropy
	Problems
6 No-Cloning Theorem, Quantum Teleportation and Spooky Correlations
	6.1 Introduction
	6.2 On Quantum Measurements
	6.3 The No-Cloning Theorem
	6.4 Quantum Teleportation
	6.5 EPR and Bell Inequalities
		6.5.1 Bertlmann's Socks
		6.5.2 Bell's Theorem
	6.6 Applications
		6.6.1 BB84 Protocol
		6.6.2 Ekert Protocol
		6.6.3 Quantum Dense Coding
	6.7 GHZ Entaglements
	Problems
7 Quantum Hardware I: Ion Trap Qubits
	7.1 Introduction
		7.1.1 The DiVincenzo Criteria
	7.2 Lagrangian and Hamiltonian Dynamics in a Nutshell
		7.2.1 Dynamics of a Translating Rotor
	7.3 Quantum Mechanics of a Free Rotor: A Poor Person's Atomic Model
		7.3.1 Rotor Dynamics and the Hadamard Gate
		7.3.2 Two-Qubit Gates
	7.4 The Cirac-Zoller Mechanism
		7.4.1 Quantum Theory of Simple Harmonic Motion
		7.4.2 A Phonon—Qubit Pair Hamiltonian
		7.4.3 Light-Induced Rotor-Phonon Interactions
	7.5 Trapped Ion Qubits
		7.5.1 Mølmer-Sørenson Coupling
	Problems
8 Quantum Hardware II: cQED and cirQED
	8.1 Introduction
	8.2 Cavity Quantum Electrodynamics (cQED)
		8.2.1 Eigenstates of the Jaynes-Cummings Hamiltonian
	8.3 Circuit QED (cirQED)
		8.3.1 Quantum upper L upper CLC Circuits
		8.3.2 Artificial Atoms
		8.3.3 Superconducting Qubits
	Problems
9 Computare Errare Est: Quantum Error Correction
	9.1 Introduction
	9.2 Quantum Error Correction
		9.2.1 Phase Flip Errors
	9.3 The Shor Code
	9.4 Stabilizers
		9.4.1 A Short Introduction to the Pauli Group
		9.4.2 Stabilizer Analysis of the Shor Code
	9.5 Stabilizers II
		9.5.1 Review of Coding Terminology
		9.5.2 Single-Qubit System
		9.5.3 Two-Qubit System
		9.5.4 Three-Qubit System
		9.5.5 The Five-Qubit Code
		9.5.6 The Seven-Qubit Steane Code
		9.5.7 Surface Codes
		9.5.8 Fault Tolerant Computing and the Threshold Theorem
10 Adiabatic Quantum Computing
	10.1 Introduction
		10.1.1 The Quantum Adiabatic Theorem
		10.1.2 The AQS Strategy
		10.1.3 Model System
		10.1.4 Avoided Crossings
		10.1.5 The Grover Algorithm
		10.1.6 Summary
	10.2 What About the Phase? Exploring Holonomic Quantum Computing
		10.2.1 The Rabi Model
		10.2.2 Geometric Phase as a Holonomy
		10.2.3 Berry's Phase
		10.2.4 Non-Abelian Phases
		10.2.5 From Phases to Forces
		10.2.6 A Topological Rabi Model
		10.2.7 Outlook
	Problems
Index