Quantum Computing A New Era of Computing

A helpful introduction to all aspects of quantum computing

Quantum computing is a field combining quantum mechanics—the physical science of nature at the scale of atoms and subatomic particles—and information science. Where ordinary computing uses bits, logical values whose position can either be 0 or 1, quantum computing is built around qubits, a fundamental unit of quantum information which can exist in a superposition of both states. As quantum computers are able to complete certain kinds of functions more accurately and efficiently than computers built on classical binary logic, quantum computing is an emerging frontier which promises to revolutionize information science and its applications.

This book provides a concise, accessible introduction to quantum computing. It begins by introducing the essentials of quantum mechanics that information and computer scientists require, before moving to detailed discussions of quantum computing in theory and practice. As quantum computing becomes an ever-greater part of the global information technology landscape, the knowledge in Quantum Computing will position readers to join a vital and highly marketable field of research and development.

The book’s readers will also find:

• Detailed diagrams and illustrations throughout
• A broadly applicable quantum algorithm that improves on the best-known classical algorithms for a wide range of problems
• In-depth discussion of essential topics including key distribution, cluster state quantum computing, superconducting qubits, and more

Quantum Computing is perfect for advanced undergraduate and graduate students in computer science, engineering, mathematics, or the physical sciences, as well as for researchers and academics at the intersection of these fields who want a concise reference.

Kuldeep Singh Kaswan, PhD, is Professor in the School of Computing Science and Engineering at Galgotias University, Greater Noida, India. He is co-editor of the upcoming Wiley title Swarm Intelligence: An Approach from Natural to Artificial.

Jagjit Singh Dhatterwal, PhD, is Associate Professor in the Department of Artificial Intelligence & Data Science at Koneru Lakshmaiah Education Foundation, Vaddeswaram, AP, India. He is co-editor of the upcoming Wiley title Swarm Intelligence: An Approach from Natural to Artificial.

Anupam Baliyan, PhD, is Professor of Engineering and Technology at Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, India.

Shalli Rani, PhD, is Professsor of Engineering and Technology at Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, India. She is co-editor of the Wiley title IoT-enabled Smart Healthcare Systems, Services and Applications.

1 INTRODUCTION OF QUANTUM COMPUTING…

1.1          Introduction

1.2          What is the exact Meaning of Quantum Computing

1.3          Origin of Quantum Computing

1.4          History of Quantum Computing

1.5          Quantum Communication

1.6          Build Quantum Computer Structure

1.7          Principle Working of Quantum Computers

1.8          Quantum Computing Use in Industry

1.9          Investors Invest Money in Quantum Technology

1.10        Applications of Quantum Computing

1.11        Quantum Computing as a Solution Technology

1.12        Conclusion

2 MERITS AND DEMERITS OF QUANTUM COMPUTING………

2.1          Introduction

2.2          Quantum as Mathematical Computation

2.3          Complexity in Quantum

2.4          Advantages and Disadvantages of the Quantum Computational Model

2.5          Additional Advantages of Quantum Computing

2.6          Additional Disadvantages of Quantum Computing

2.7          Hybridization of Quantum Computing

2.8          Structure of QRAM

2.9          Algorithm in Quantum Computing

2.9.1      Designing of Quantum Algorithm

2.10           Alteration in Quantum Blocks

3              TOOLS ANF TECHNIQUES ON QUANTUM THEORY.…

3.1          Classical Information

3.2          Information Content in a Signal

3.3          Theory of Information and Entropy

3.4          Probability Basic

3.5          The No-Cloning Theorem

3.6          Trace Distance

3.7          Fidelity

3.8          Entanglement Of Formation and Concurrence

3.9          Information Content and Entropy

4 BUILDING BLOCKS OF QUANTUM COMPUTING……...

4.1          Single-Qubit

4.1.1 The Quantum Mechanics of Photon Polarization

4.2          Multi-Qubit

4.2.1      Quantum State Spaces

4.2.2      Direct Sums of Vector Spaces

4.2.3      Tensor Products of Vector Spaces

4.2.4      The state spaces of an n-Qubit System

4.2.5      Entangled States

4.2.6      Basics of Multi-Qubit Measurement

4.3          Measurement of Multi-Qubit

4.3.1      Notation for Mathematical Operations Using the Bra/Ket System Developed by Dirac

4.3.2      Projection Operators for Measurement

4.3.3      The Measurement Postulate

4.3.4      EPR Paradox and Bell’s Theorem

4.3.5      Setup for Bell’s Theorem

4.3.6      What Quantum Mechanics Predicts

4.3.7      Applying Bell's Theorem to the Predictions of Any Local Theory of Hidden Variables

4.3.8      Bell’s Inequality

4.4          State of Quantum Transformation

4.4.1      Single-step transformations

4.4.2      Impossible Transformations: The No-Cloning Principle

4.4.3      The Pauli Transformations

4.4.4      The Hadamard Transformation

4.4.5      Multiple-Qubit Transformations from Single-Qubit Transformations

4.4.6      The Controlled-NOT and Other Singly Controlled Gates

4.4.7      Dense Coding

4.4.8      Classical Bits in Dense Coding

4.4.9      Quantum Teleportation

4.4.10    Realizing Unitary Transformations as Quantum Circuits

4.4.11    Decomposition of Single-Qubit Transformations

4.4.12    Singly Controlled Single-Qubit Transformations

4.4.13    Multiply Controlled Single-Qubit Transformations

4.4.14    General Unitary Transformations

4.4.15    A Universally Approximating Set of Gates

4.4.16    The Standard Circuit Model

5 ALGORITHM STRUCTURE OF QUANTUM COMPUTING……

5.1          Introduction

5.2          Quantum Algorithm

5.3          Quantum Computing Rule 1

5.4          Quantum Computing Rule 2

5.5          Quantum Computing Rule 3

5.6          Quantum Computing Rule 4

5.7          Quantum Computing Rule 5

5.8          Quantum Computing Rule 6

5.9          Quantum Computing Rule 7

5.10        Quantum Computing Rule 8

6              ALGORITHM OF AMPLITUDE AMPLIFICATION …………………….

6.1          Introduction

6.2          Availability Bias

6.3          Amplitude Amplification Algorithm

6.3.1 Mathematical details for the amplitude amplification algorithm

6.4          Quantum Amplitude Estimation and Quantum Counting

6.5          The quantum amplitude estimation algorithm

6.5.1      Mathematical description of amplitude estimation algorithm

6.6          The Quantum Counting Algorithm

6.7          Searching Without Knowing the Success Probability

7              ERROR CORRECTION CODE IN QUANTUM NOISE …………

7.1          Introduction

7.2          Basic forms of Error-Correcting Code in Quantum Technologies

7.1.1 Single Bit-Flip Errors in Quantum Computing

        7.1.2 Single Qubit Coding in Quantum Computing

        7.1.3 Error Correcting Code in Quantum Technology

7.3 Framework for Quantum Error Correcting Codes

 7.3.1 Traditional based on Error Correcting Codes

 7.3.2 Quantum Error Decode Mechanisms

 7.3.3 Correction Sets in Quantum Coding Error

 7.3.4 Quantum Errors Detection

 7.3.5 Basic Knowledge Representation of Error Correcting Code

 7.3.6 Quantum codes as a tool for error detection and correction

 7.3.7 Quantum Error Correction Across Multiple Blocks

 7.3.8 Computing on Encoded Quantum States

 7.3.9 Using Linear Transformation of Correctable Codes

 7.3.10 The Classical Independent Error Model

 7.3.11 Quantum Independent Error Models

7.4 CSS Codes

7.4.1 Dual Classical Codes

7.4.2 CSS Codes Built Using Traditional Codes That Meet a Duality Consequence

7.4.3 The Steane Code

7.5 Stabilizer Codes

7.5.1 Binary Observables for Quantum Error Correction

7.5.2 Error-Correcting Pauli Observables in Quantum Systems

7.5.3 Correcting Errors using Stabilizer Code

7.5.4 Computing on the Encoded Stabilizer States

7.6 CSS Codes as Stabilizer Codes

8 FAULT TOLERANCE IN QUANTUM COMPUTING

 8.1 Introduction

8.2 Setting the Stage for Robust Quantum Computation

8.3 Fault-Tolerant Computation Using Steane’s Code

8.3.1 The Problem with Syndrome Computation

8.3.2 Fault-Tolerant Syndrome Extraction and Error Correction

8.3.3 Fault-Tolerant Gates for Steane’s Code

8.3.4 Fault-Tolerant Measurement

8.3.5 Fault-Tolerant State Preparation of |π/4

8.4 Robust Quantum Computation

8.4.1 Concatenated Coding

8.4.2 A Threshold Theorem

9 CRYPTOGRAPHY IN QUANTUM COMPUTING

9.1 Introduction of RSA Encryption

9.2 A Brief Overview of RSA Encryption

9.3 Basic Quantum Cryptography

9.4 An Example Attack: The Controlled Not Attack

9.5 The B92 Protocol

9.6 The E91 Protocol

10 BUILDING CLUSTERS IN QUANTUM COMPUTING

10.1 Introduction

10.1.1 Cluster States

10.2 Cluster States Preparation

10.3 Adjacency Matrices

10.4 Stabilizer States

10.4.1 Aside: Entanglement Witness

10.5 Cluster State Processing

11 ADVANCE QUANTUM COMPUTING

11.1 Introduction

11.2 Computing with Superpositions

    11.2.1 The Walsh-Hadamard Transformation

      11.2.2 Quantum Parallelism

11.3 Notions of Complexity

11.3.1 Query Complexity

11.3.2 Communication Complexity

11.4 A Simple Quantum Algorithm

11.4.1 Deutsch’s Problem

11.5 Quantum Subroutines

11.5.1 The Importance of Unentangling Temporary Qubits in Quantum Subroutines

11.5.2 Phase Change for a Subset of Basis Vectors

11.5.3 State-Dependent Phase Shifts

11.5.4 State-Dependent Single-Qubit Amplitude Shifts

11.6 A Few Simple Quantum Algorithms

11.6.1 Deutsch-Jozsa Problem

11.6.2 Bernstein-Vazirani Problem

11.6.3 Simon’s Problem

11.6.4 Distributed Computation

11.7 Comments on Quantum Parallelism

11.8 Machine Models and Complexity Classes

11.8.1 Complexity Classes

11.8.2 Complexity: Known Results

11.9 Quantum Fourier Transformations

11.9.1 The Classical Fourier Transform

11.9.2 The Quantum Fourier Transform

11.9.3 A Quantum Circuit for Fast Fourier Transform

11.10 Shor’s Algorithm

11.10.1 The Quantum Core

11.10.2 Classical Extraction of the Period from the Measured Value

11.10.3 The Efficiency of Shor’s Algorithm

11.11 Omitting the Internal Measurement

11.12 Generalizations

11.13 Grover’s Algorithm use Solve sets of problems

11.13.1 Outline method of superposition

11.13.2 Setup of Black Box

11.13.3 The Iteration Step

11.13.4 How Many Iterations?

11.14 Good State Functions

11.14.1 The Two-Dimensional Geometry

11.15 Optimality of Grover’s Algorithm

11.15.1 Reduction to Three Inequalities

11.16 DE randomization of Grover’s Algorithm and Amplitude Amplification

11.16.1 Approach 1: Modifying Each Step

11.16.2 Approach 2: Modifying Only the Last Step

11.16.3 Unknown Number of Solutions

11.16.4 Varying the Number of Iterations

11.16.5 Quantum Counting

12 APPLICATIONS OF QUANTUM COMPUTING

12.1 Introduction

12.2 Teleportation Steps

12.3 The Peres Partial Transposition Condition

12.4 Entanglement Swapping

12.5 Superdense Coding

Index

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.