A Quantum Computation Workbook

Preface
Contents
1 The Postulates of Quantum Mechanics
	1.1 Quantum States
		1.1.1 Pure States
		1.1.2 Mixed States
	1.2 Time Evolution of Quantum States
		1.2.1 Unitary Dynamics
		1.2.2 Quantum Noisy Dynamics
	1.3 Measurements on Quantum States
		1.3.1 Projection Measurements
		1.3.2 Generalized Measurements
2 Quantum Computation: Overview
	2.1 Single-Qubit Gates
		2.1.1 Pauli Gates
		2.1.2 Hadamard Gate
		2.1.3 Rotations
	2.2 Two-Qubit Gates
		2.2.1 CNOT, CZ, and SWAP
		2.2.2 Controlled-Unitary Gate
		2.2.3 General Unitary Gate
	2.3 Multi-Qubit Controlled Gates
		2.3.1 Gray Code
		2.3.2 Multi-Qubit Controlled-NOT
	2.4 Universal Quantum Computation
	2.5 Measurements
3 Realizations of Quantum Computers
	3.1 Quantum Bits
	3.2 Dynamical Scheme
		3.2.1 Implementation of Single-Qubit Gates
		3.2.2 Implementation of CNOT
	3.3 Geometric/Topological Scheme
		3.3.1 A Toy Model
		3.3.2 Geometric Phase
	3.4 Measurement-Based Scheme
		3.4.1 Single-Qubit Rotations
		3.4.2 CNOT Gate
		3.4.3 Graph States
4 Quantum Algorithms
	4.1 Quantum Teleportation
		4.1.1 Nonlocality in Entanglement
		4.1.2 Implementation of Quantum Teleportation
	4.2 Deutsch-Jozsa Algorithm & Variants
		4.2.1 Quantum Oracle
		4.2.2 Deutsch-Jozsa Algorithm
		4.2.3 Bernstein-Vazirani Algorithm
		4.2.4 Simon's Algorithm
	4.3 Quantum Fourier Transform (QFT)
		4.3.1 Definition and Physical Meaning
		4.3.2 Quantum Implementation
		4.3.3 Semiclassical Implementation
	4.4 Quantum Phase Estimation (QPE)
		4.4.1 Definition
		4.4.2 Implementation
		4.4.3 Accuracy
		4.4.4 Simulation of the von Neumann Measurement
	4.5 Applications
		4.5.1 The Period-Finding Algorithm
		4.5.2 The Order-Finding Algorithm
		4.5.3 Quantum Factorization Algorithm
		4.5.4 Quantum Search Algorithm
5 Quantum Decoherence
	5.1 How Does Decoherence Occur?
	5.2 Quantum Operations
		5.2.1 Kraus Representation
		5.2.2 Choi Isomorphism
		5.2.3 Unitary Representation
		5.2.4 Examples
	5.3 Measurements as Quantum Operations
	5.4 Quantum Master Equation
		5.4.1 Derivation
		5.4.2 Examples
		5.4.3 Solution Methods
		5.4.4 Examples Revisited
	5.5 Distance Between Quantum States
		5.5.1 Norms and Distances
		5.5.2 Hilbert–Schmidt and Trace Norms
		5.5.3 Hilbert–Schmidt and Trace Distances
		5.5.4 Fidelity
6 Quantum Error-Correction Codes
	6.1 Elementary Examples: Nine-Qubit Code
		6.1.1 Bit-Flip Errors
		6.1.2 Phase-Flip Error
		6.1.3 Shor's Nine-Qubit Code
	6.2 Quantum Error Correction
		6.2.1 Quantum Error-Correction Conditions
		6.2.2 Discretization of Errors
	6.3 Stabilizer Formalism
		6.3.1 Pauli Group
		6.3.2 Properties of the Stabilizers
		6.3.3 Unitary Gates in Stabilizer Formalism
		6.3.4 Clifford Group
		6.3.5 Measurements in Stabilizer Formalism
		6.3.6 Examples
	6.4 Stabilizer Codes
		6.4.1 Bit-Flip Code
		6.4.2 Phase-Flip Code
		6.4.3 Nine-Qubit Code
		6.4.4 Five-Qubit Code
	6.5 Surface Codes
		6.5.1 Toric Codes
		6.5.2 Planar Codes
		6.5.3 Recovery Procedure
7 Quantum Information Theory
	7.1 Shannon Entropy
		7.1.1 Definition
		7.1.2 Relative Entropy
		7.1.3 Mutual Information
		7.1.4 Data Compression
	7.2 Von Neumann Entropy
		7.2.1 Definition
		7.2.2 Relative Entropy
		7.2.3 Quantum Mutual Information
	7.3 Entanglement and Entropy
		7.3.1 What is Entanglement?
		7.3.2 Separability Tests
		7.3.3 Entanglement Distillation
		7.3.4 Entanglement Measures
Appendix A Linear Algebra
A.1  Vectors
A.1.1  Vector Space
A.1.2  Hermitian Product
A.1.3  Basis
A.1.4  Representations
A.2  Linear Operators
A.2.1  Linear Maps
A.2.2  Representations
A.2.3  Hermitian Conjugate of Operators
A.3  Dirac's Bra-Ket Notation
A.4  Spectral Theorems
A.4.1  Spectral Decomposition
A.4.2  Functions of Operators
A.5  Factorization of Operators
A.6  Tensor-Product Spaces
A.6.1  Vectors in a Product Space
A.6.2  Operators on a Product Space
Appendix B Superoperators
B.1  Operators as Vectors
B.2  Superoperators
B.2.1  Matrix Representation
B.2.2  Operator-Sum Representation
B.2.3  Choi Isomorphism
B.3  Partial Trace
B.4  Partial Transposition
Appendix C Group Theory
C.1  The Concept
C.2  Classes
C.3  Invariant Subgroups
C.4  Cosets and Quotient Groups
C.5  Product Groups
Appendix D Mathematica Application Q3
D.1  Installation
D.2  Quick Start
Appendix E Integrated Compilation of Demonstrations
E.1  Installation
E.2  Quick Start
Appendix F Solutions to Select Problems
F.1  The Postulates of Quantum Mechanics
F.2  Quantum Computation: Overview
F.3  Quantum Computers
F.4  Quantum Algorithms
F.5  Decoherence
F.6  Quantum Error-Correction Codes
F.7  Quantum Information Theory
Appendix  Bibliography
Index