1st Edition

# Quantum Computing From Linear Algebra to Physical Realizations

**Also available as eBook on:**

Covering both theory and progressive experiments, Quantum Computing: From Linear Algebra to Physical Realizations explains how and why superposition and entanglement provide the enormous computational power in quantum computing. This self-contained, classroom-tested book is divided into two sections, with the first devoted to the theoretical aspects of quantum computing and the second focused on several candidates of a working quantum computer, evaluating them according to the DiVincenzo criteria.

Topics in Part I

- Linear algebra
- Principles of quantum mechanics
- Qubit and the first application of quantum information processing—quantum key distribution
- Quantum gates
- Simple yet elucidating examples of quantum algorithms
- Quantum circuits that implement integral transforms
- Practical quantum algorithms, including Grover’s database search algorithm and Shor’s factorization algorithm
- The disturbing issue of decoherence
- Important examples of quantum error-correcting codes (QECC)

Topics in Part II

- DiVincenzo criteria, which are the standards a physical system must satisfy to be a candidate as a working quantum computer
- Liquid state NMR, one of the well-understood physical systems
- Ionic and atomic qubits
- Several types of Josephson junction qubits
- The quantum dots realization of qubits

Looking at the ways in which quantum computing can become reality, this book delves into enough theoretical background and experimental research to support a thorough understanding of this promising field.

**From linear algebra to quantum computing**

**Basics of Vectors and Matrices**

Vector Spaces

Linear Dependence and Independence of Vectors

Dual Vector Spaces

Basis, Projection Operator, and Completeness Relation

Linear Operators and Matrices

Eigenvalue Problems

Pauli Matrices

Spectral Decomposition

Singular Value Decomposition (SVD)

Tensor Product (Kronecker Product)

**Framework of Quantum Mechanics**

Fundamental Postulates

Some Examples

Multipartite System, Tensor Product, and Entangled State

Mixed States and Density Matrices

**Qubits and Quantum Key Distribution**

Qubits

Quantum Key Distribution (BB84 Protocol)

**Quantum Gates, Quantum Circuit, and Quantum Computer**

Introduction

Quantum Gates

Correspondence with Classical Logic Gates

No-Cloning Theorem

Dense Coding and Quantum Teleportation

Universal Quantum Gates

Quantum Parallelism and Entanglement

**Simple Quantum Algorithms**

Deutsch Algorithm

Deutsch–Jozsa Algorithm and Bernstein–Vazirani Algorithm

Simon’s Algorithm

**Quantum Integral Transforms**

Quantum Integral Transforms

Quantum Fourier Transform (QFT)

Application of QFT: Period-Finding

Implementation of QFT

Walsh–Hadamard Transform

Selective Phase Rotation Transform

**Grover’s Search Algorithm**

Searching for a Single File

Searching for

*d*Files

**Shor’s Factorization Algorithm**

The RSA Cryptosystem

Factorization Algorithm

Quantum Part of Shor’s Algorithm

Probability Distribution

Continued Fractions and Order-Finding

Modular Exponential Function

**Decoherence**

Open Quantum System

Measurements as Quantum Operations

Examples

Lindblad Equation

**Quantum Error-Correcting Codes (QECC)**

Introduction

3-Qubit Bit-Flip Code and Phase-Flip Code

Shor’s 9-Qubit Code

Calderbank–Shor–Steane (CSS) 7-Qubit QECC

DiVincenzo–Shor 5-Qubit QECC

**Physical realizations of quantum computing**

**DiVincenzo Criteria**

Introduction

DiVincenzo Criteria

Physical Realizations

Beyond DiVincenzo Criteria

**NMR Quantum Computer**

Introduction

NMR Spectrometer

Hamiltonian

Implementation of Gates and Algorithms

Time-Optimal Control of NMR Quantum Computer

Measurements

Preparation of Pseudopure State

DiVincenzo Criteria

**Trapped Ions**

Introduction

Electronic States of Ion as Qubit

Ions in Paul Trap

Ion Qubit

Quantum Gates

Readout

DiVincenzo Criteria

**Quantum Computing with Neutral Atoms**

Introduction

Trapping Neutral Atoms

1-Qubit Gate

Quantum State Engineering of Neutral Atoms

Preparation of Entangled Neutral Atoms

DiVincenzo Criteria

**Josephson Junction Qubits**

Introduction

Nanoscale Josephson Junctions and SQUIDs

Charge Qubit

Flux Qubit

Quantronium

Current-Biased Qubit

Readout

Coupled Qubits

DiVincenzo Criteria

**Quantum Computing with Quantum Dots**

Introduction

Mesoscopic Semiconductors

Electron Charge Qubit

Electron Spin Qubit

DiVincenzo Criteria

**Appendix: Solutions to Selected Exercises**

Index

### Biography

Mikio Nakahara, Tetsuo Ohmi

The book is very well structured and offers good theoretical explanations reinforced by examples. As the authors mention in the Preface, the book can be used for a quantum computing course. It is also recommended to advanced undergraduate students, postgraduate students and researchers in physics, mathematics and computer science.

—Zentralblatt MATH1185