Quantum Computing Explained

CONTENTS

Preface

1    A BRIEF INTRODUCTION TO INFORMATION THEORY

Classical Information

Information Content in a Signal

Entropy and Shannon’s Information Theory

Probability Basics

Example 1.1

Solution

Exercises

2    QUBITS AND QUANTUM STATES

The Qubit

Example 2.1

Solution

Vector Spaces

Example 2.2

Solution

Linear Combinations of Vectors

Example 2.3

Solution

Uniqueness of a Spanning Set

Basis and Dimension

Inner Products

Example 2.4

Solution

Example 2.5

Solution

Orthonormality

Gram-Schmidt Orthogonalization

Example 2.6

Solution

Bra-Ket Formalism

Example 2.7

Solution

The Cauchy-Schwartz and Triangle Inequalities

Example 2.8

Solution

Example 2.9

Solution

Summary

Exercises

3    MATRICES AND OPERATORS

Observables

The Pauli Operators

Outer Products

Example 3.1

Solution

You Try It

The Closure Relation

Representations of Operators Using Matrices

Outer Products and Matrix Representations

You Try It

Matrix Representation of Operators in Two-Dimensional Spaces

Example 3.2

Solution

You Try It

Definition: The Pauli Matrices

Example 3.3

Solution

Hermitian, Unitary, and Normal Operators

Example 3.4

Solution

You Try It

Definition: Hermitian Operator

Definition: Unitary Operator

Definition: Normal Operator

Eigenvalues and Eigenvectors

The Characteristic Equation

Example 3.5

Solution

You Try It

Example 3.6

Solution

Spectral Decomposition

Example 3.7

Solution

The Trace of an Operator

Example 3.8

Solution

Example 3.9

Solution

Important Properties of the Trace

Example 3.10

Solution

Example 3.11

Solution ...