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THE IMPLEMENTATION OF CONTROLLED ADIABATIC AND NONADIABATIC EVOLUTIONS

QUANTUM GATES IN OPEN SYSTEMS

BY

ABDERRAHIM BENMACHICHE

A thesis submitted in fulfilment of the requirement for the degree of Doctor of Philosophy in Computer Science

Kulliyyah of Information and Communication Technology International Islamic University Malaysia

November 2020

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ABSTRACT

In the last two decades, theoretical and experimental works on quantum information have achieved noticeable progress. The theoretical results demonstrate that quantum computers can perform computational tasks better than the classical one. However, building a complete quantum computer that can solve real computational problems is still difficult. The main challenge to implement a quantum computer is to come with a model that is very robust to the environment effect. Today, the two fundamental quantum computational models frequently used in the experimental side are the gate model and the adiabatic evolution one. These two models are basically used to implement quantum gates and quantum circuits. Recently, Itay Hen could combine the gate model and the adiabatic one to come with a new model called the controlled adiabatic evolutions. In this thesis, we investigated the robustness of Itay’s model to the external environment. Besides, we proposed a new model similar to Itay’s one that gives a speed advantage to implement quantum gates. We start our work by illustrating the general frame of this research study. After, we provided an overview of the area of quantum computation. We discussed the origins and the history of the field, and the primary application of quantum computing. In the main work, we introduced Itay’s model and illustrated how we could implement the fundamental elementary gates and prepare Bell’s states based on the controlled adiabatic evolutions model. Next, we studied the robustness of Itay’s model to the different type of noise using Monte-Carlo quantum trajectory and Lindblad master equation. As an alternative to Itay’s model, we proposed a new approach similar to his model but with nonadiabatic evolutions. The new model allows speeding the implementation process of the quantum gate. Also, we studied the reliability of our model in the case of the presence of noise using the same approaches used with Itay’s model. We concluded the study by comparing the effectiveness of the controlled adiabatic process with the controlled nonadiabatic one.

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ثحبلا ةصلاخ

ABSTRACT IN ARABIC

,ينيضالما نيدقعلا للاخ ةيبيرجتلاو ةيرظنلا لامعلأا تققح

ملع في امدقت مكلا تامولعم

.اظوحلم

أ ثيح رهظ ت نأ ةيرظنلا جئاتنلا بيساولحا

لكشب ةيباسلحا ماهلما ءادأ اهنكيم ةيمكلا

لحا نم لضفأ بيساو

.ةيكيسلاكلا

عيضاولما دحأ رثكلأا

مامتها ا في لامج لحا ساو يمكلا ب ة

ايلاح يرثتأ ةسارد وه ةيجرالخا لماوعلا

ىلع فلتمخ ولحا جذانم

اس ي ةيمكلا ب

ةدوجولما .

نكل

يمك بساح ةعانص

يلك .ايصعتسم ارمأ لازي لا

بيساولحا ةعانص هجاوت تيللا تيادحتلا مهأ

يه ةيمكلا .ةيجرالخا لماوعلا هاتجا ينتم جضونم ءاشنا

اذه انموي في

لا مهأ امن جذ في ةلمعتسلما

بييرجتلا لالمجا و تاقللحا جذونم اهم

جذونم تباثلا مدقتلا .

نيذه مادختسا متي ، امومع

ينجذومنلا لإ

ءاشن لا .يمكلا بوساحلل ةيساسلاا رصانع ،ةيرخلأا ةنولآا في

عاطتسا ينه ياتيإ

جذونم جمدب تاقللحا

و تباثلا مدقتلا جذونم

ىمسي ديدج جذونم يمدقتل

تباثلا مدقتلا هيجوت .

انمق ، ةحورطلأا هذه في

ىدم ةساردب ةمواقم

جذونم ينه يتاإ

عاونأ فلتخلم ا

ةيمكلا شيوشتل .

ج اًجذونم انحترقا امك جذومنل اًبهاشم اًديد

يتاإ ي موق عيرستب تاقللحا زانجإ ةدم .

أدبن في

راطلإا حيضوتب اذه انلمع

هذله ماعلا ةحورطلأا

. دعب اه ن ، تب موق دق يم لامج نع ةماع ةلمح

لحا اس و مكلا ب ي

. اهيف شقانن ثيح

ئدابلما قيبطتلاو تا يساسلأا ة

وحلل اس ي .ةيمكلا ب

في

انلمع لأا يساس ، جونم انحرش إ

و ياتي و حض ان هيف اننكيم فيك إ

ةيساسلأا تاقللحا زانج و

دادعإ تادحو ليب

ا متع ادا ىلع جذونم تباثلا مدقتلا هيجوت .

كلذك

، انمق ب انسرد ةمواقم

إ جضونم ي

يات فلتخلم عاونأ

ةيمكلا تاشيوشتلا

م مادختسبا تيلداع

تينوم .دلابدنيل و ولراك

جذومنل ليدبك يإت

يا

، انمق با انحترق هباشم جضونم

تبثا مدقت نودب نكل هيجوتلا ىلع دمتعي .

انجذونم ديدلجا حمسي عيرستب ةيلمع

تاقللحا زانجا ، اًضيأ .

ةساردب انمق

ةمواقم ةدش

جذومنلا

يذلا هانحرط عاونأ فلتخلم

ةيمكلا تاشوشتلا ا سفن مادختسبا

عم اهانمدختسا تيلا ةقيرطل

ياتيإ جذونم .

ذه انثبح متنخو ياتيأ هرط يذلا جذومنلا عم هانحرط يذلا جذومنلا ةنراقبم ا

.

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APPROVAL PAGE

The thesis of Abderrahim Benmachiche has been approved by the following:

_____________________________

Mohamed Ridza Wahiddin Supervisor

_____________________________

Azeddine Messikh Co-Supervisor

_____________________________

Bakhram Umarov Internal Examiner

_____________________________

Zuriati Ahmed Zukarnain External Examiner

_____________________________

Smail Bougouffa External Examiner

_____________________________

Gairuzami Bin Mat Ghani Chairman

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DECLARATION

I hereby declare that this thesis is the result of my own investigations, except where otherwise stated. I also declare that it has not been previously or concurrently submitted as a whole for any other degrees at IIUM or other institutions.

Abderrahim Benmachiche

Signature ... Date ...

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COPYRIGHT

INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA

DECLARATION OF COPYRIGHT AND AFFIRMATION OF FAIR USE OF UNPUBLISHED RESEARCH

THE CONTROLLED ADIABATIC AND NONADIABATIC EVOLUTIONS QUANTUM GATES

I declare that the copyright holders of this thesis are jointly owned by the student and IIUM.

Copyright © 2020 Abderrahim Benmachiche and International Islamic University Malaysia. All rights reserved.

No part of this unpublished research may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without prior written permission of the copyright holder except as provided below

1. Any material contained in or derived from this unpublished research may only be used by others in their writing with due

acknowledgement.

2. IIUM or its library will have the right to make and transmit copies (print or electronic) for institutional and academic purposes.

3. The IIUM library will have the right to make, store in a retrieved system and supply copies of this unpublished research if requested by other universities and research libraries.

By signing this form, I acknowledged that I have read and understand the IIUM Intellectual Property Right and Commercialization policy.

Affirmed by Abderrahim Benmachiche

……..……….. ………..

Signature Date

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ACKNOWLEDGEMENTS

Firstly, it is my utmost pleasure to dedicate this work to my dear parents and my family, who granted me the gift of their unwavering belief in my ability to accomplish this goal:

thank you for your support and patience.

A special thanks to my two supervisors Dr. Azeddine Messikh and Prof.

Mohamed Ridza Wahiddin. For their continuous support, encouragement, and leadership, and for that, I will be forever grateful.

Besides, I would say thanks to the Malaysian Education Ministry for their financial support under research grant number FRGS17-024-0590.

Finally, I wish to express my appreciation and thanks to those who provided their time, effort, and support for this work. To the members of my thesis committee, thank you for the support.

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TABLE OF CONTENTS

Abstract ... ii

Abstract in Arabic ... iii

Approval Page ... iv

Declaration ... v

Copyright ... vi

Acknowledgements ... vii

List of Tables ... xi

List of Figures ... xii

CHAPTER ONE: INTRODUCTION ... 1

1.1 Background of the Study ... 1

1.2 Statement of the Problem ... 3

1.3 Research Questions ... 5

1.4 Research Objectives ... 5

1.5 Significance of the Study ... 6

1.6 Organization of the Thesis ... 6

CHAPTER TWO: LITERATURE REVIEW ... 8

2.1 Introduction ... 8

2.2 Quantum Mechanics... 10

2.3 Qubits ... 11

2.3.1 Bra-ket Notation ... 11

2.3.2 The Bloch Sphere ... 12

2.3.3 The physical Implementation of a Qubit ... 13

2.3.4 Measurement ... 15

2.3.5 Multiple Qubits ... 15

2.4 Quantum Gates ... 16

2.4.1 No-Cloning Theorem ... 17

2.4.2 Entanglement ... 18

2.5 Quantum Computing Application ... 18

2.5.1 Shor’s Quantum Factorisation Algorithm ... 19

2.5.2 Quantum Key Distribution ... 20

2.5.3 Teleportation ... 20

2.6 Divincenzo's Criteria ... 22

2.7 Related Work ... 22

2.8 Further Reading ... 31

2.9 Summary ... 33

CHAPTER THREE: THE CONTROLLED ADIABATIC EVOLUTIONS GATES ... 34

3.1 Introduction ... 34

3.2 Controlled Adiabatic Evolutions ... 35

3.3 Single-Qubit Quantum Gates ... 39

3.3.1 NOT Gate ... 39

3.3.2 Flip Gate ... 41

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3.3.3 Pauli-Y Gate ... 42

3.3.4 Hadamard Gate ... 43

3.3.5 Phase Gate ... 44

3.3.6 π8 Phase Gate ... 45

3.4 Two-Qubit Quantum Gates ... 46

3.4.1 Controlled-NOT Gate ... 46

3.4.2 Controlled-Z Gate ... 47

3.4.3 Controlled-Phase Gate ... 48

3.5 Three-Qubit Quantum Gates ... 50

3.5.1 Toffoli Gate ... 50

3.6 Bell’s States ... 51

3.6.1 |Φ+ 〉 = 1 √2(|00〉 + |11〉) ... 52

3.6.2 |Ψ+ 〉 = 1 √2(|01〉 + |10〉) ... 52

3.6.3 |Ψ- 〉 = 1 √2(|01〉 - |10〉) ... 52

3.6.4 |Φ+ 〉 = 1 √2(|00〉 - |11〉) ... 53

3.7 Summary ... 53

CHAPTER FOUR: SINGLE GATES WITH CONTROLLED ADIABATIC EVOLUTIONS IN OPEN SYSTEMS ... 55

4.1 Introduction ... 55

4.2 Open Systems ... 57

4.3 Fidelity ... 61

4.3.1 The Flip Gate ... 63

4.3.2 NOT Gate ... 64

4.3.3 Pauli-Y Gate ... 65

4.3.4 Hadamard Gate ... 66

4.3.5 Bell’s States ... 67

4.4 Conclusion ... 68

CHAPTER FIVE: QUANTUM ROTATION GATES WITH CONTROLLED NONADIABATIC EVOLUTIONS ... 70

5.1 Introduction ... 70

5.2 Nonadiabatic Holonomic Gates By A Single-Shot Implementation... 71

5.3 Controlled Nonadiabatic Evolutions ... 73

5.4 Open System ... 74

5.4.1 Flip Gate ... 75

5.4.2 NOT Gate ... 76

5.4.3 Pauli-Y Gate ... 77

5.4.4 Hadamard Gate ... 78

5.5 Summary ... 80

CHAPTER SIX: CONCLUSIONS AND FUTURE WORK ... 81

6.1 Introduction ... 81

6.2 Conclusion of Research... 81

6.3 Future Work Recommendation ... 83

REFERENCES ... 86

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APPENDIX A: SCHRÖDINGER'S EQUATION ... 93

APPENDIX B: QUANTUM ADIABATIC PROCESS ... 96

APPENDIX C: QUANTUM MONTE-CARLO EQUATION………..99

APPENDIX D: LINDBLAD MASTER EQUATION………...101

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LIST OF TABLES

Table No. Page No.

2.1 Bobs Conditional Operator in Teleportation Protocol 21

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LIST OF FIGURES

Figure No. Page No.

2.1 Illustration the State of the Qubit |𝜓⟩ = cos𝜃

2|0⟩ + 𝑒𝑖𝜙 sin𝜃

2|1⟩, as

a Vector on the Block Sphere 13

2.2 Different Physical Illustration of a Qubit Using Atomic Orbitals,

Photon Polarization and the Spin of the Electron. 14 3.1 The Dashed and Solid Slack Lines Represent the Adiabatic

Evolutions of the Auxiliary Qubit According to the Ground States of

the Hamiltonian 𝐻0, 𝐻𝜙 Respectively 39

4.1 The Blue Dashed Line Show that the Relative Phase Vanishes in the Closed System, whereas the Red Line Illustrates the Change of the Relative Phase when we Introduced the Dephasing of the Ground

State 𝐶00. 61

4.2 The Blue Dashed Line Shows that the Relative Phase Vanishes in the Closed System, whereas the Red Line Illustrates the Change of the Relative Phase when we Introduced the Dephasing Operator

Corresponding to the Collision Operator 𝐶01 61

4.3 Generating 1000 Random States Uniformly Distributed on the Bloch

Sphere 62

4.4 The Fidelity of the Flip Gate in the Presence of Noise 64 4.5 The Fidelity of the NOT Gate in the Presence of Noise 65 4.6 The Fidelity of the Pauli-Y Gate in the Presence of Noise 66 4.7 The Fidelity of the Hadamard Gate in the Presence of Noise 67 4.8 The Fidelity of the Bell’s States in the Presence of Noise 68 5.1 The Fidelity of the Flip Gate in the Presence of Noise 76 5.2 The Fidelity of the NOT Gate in the Presence of Noise 77 5.3 The Fidelity of the Pauli-Y Gate in the Presence of Noise 78 5.4 The Fidelity of the Hadamard Gate in the Presence of Noise 79

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CHAPTER ONE INTRODUCTION

1.1 BACKGROUND OF THE STUDY

Nowadays, many important problems exist that are intractable or impossible to solve using classical computers, such as, the factorisation of the semi-prime numbers. The factorisation of a number composed of the product of two prime numbers of length equals to two-hundreds digits may take several decades to the fastest classical computer in the world. In 1994, P. Shor from MIT proposed a quantum algorithm that can solve the problem of factorisation in polynomial time (P. W. Shor, 1994; P.

W. Shor, 1999). Later, other quantum algorithms have been introduced to solve difficult computational problems that classical computers unable to do; for instance, Simon’s quantum algorithm can fix the black box problem, and Grover’s quantum algorithm can optimise the searching time in the unstructured database (Grover, 1996, 1997; Simon, 1997).

The high efficiency and speed that quantum algorithms provide compared to classical one prompt scientists to seek for a new generation of computers called quantum computers. This new generation has a unique hardware type with a computational mechanism based on quantum mechanics hypotheses. The basic information of quantum computers called qubit. The qubit is based on the superposition states in which the states 0 and 1 are overlapped. To manipulate qubits, we need to construct a new type of circuits called quantum circuits or quantum gates.

Many different approaches have been proposed to develop these kinds of circuits.

The most common, and the one that several research efforts are focusing on, is the adiabatic quantum evolution. This approach was introduced in quantum computing

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by E. Farhi et al.(E. Farhi et al., 2001; Edward Farhi, Goldstone, Gutmann, & Sipser, 2000). Its concept is based on the adiabatic evolution theorem where the quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough (Albash & Lidar, 2018; Born & Fock, 1928; Kato, 1950; Messiah, 1961).

Many protocols have been introduced to implement quantum gates based on the adiabatic evolution approach. For instance, X. Lacour et al. lead an experiment to construct the Hadamard, SWAP and CNOT quantum gates using fixed atoms in an optical cavity scheme (X. Lacour, S. Guerin, N. V. Vitanov, L. P. Yatsenko, & H.

R. Jauslin, 2006; Sangouard, Lacour, Guerin, & Jauslin, 2005, 2006). These gates work perfectly in a closed system. However, in an open system, their performance becomes less effective as Y. H. Issoufa and A. Messikh. demonstrated for the case of Hadamard gate (Issoufa & Messikh, 2010). The closed system is an ideal system where the qubits are not affected by the environment, whereas, the open system is the realistic one where the quantum states interact with the environment, and they are susceptible to different dephasing types (dissipative effects). Thus, it is essential to mention that the reliability of quantum circuits depends on its robustness to the different dissipative impact.

Recently, I. Hen introduced a new method to implement quantum adiabatic gates. Itay used quantum adiabatic evolution somewhat unconventionally. In his approach, he considered the adiabatic evolution of several systems in parallel. This technique enabled him to construct two models of quantum gates which are the adiabatic single-qubit rotation gates and the adiabatic controlled-rotation gates. Itay claims that the combination of the two models gives a universal set and may, therefore, be used to build complicated circuits blocks (Hen, 2015).

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In this thesis, we demonstrate that the controlled adiabatic evolutions model proposed by Itay can be used to implement a universal set of quantum gates and prepare the four Bell’s states (Bell, 1964). Next, we extend and study the robustness of the controlled adiabatic evolutions in an open system (Benmachiche, Bahloul, Mahmoud, & Messikh, 2018). In the end, we propose a counterpart protocol to Itay’s approach based on nonadiabatic evolutions and study the robustness of this new protocol in an open system (Abdelrahim, Benmachiche, Mahmoud, & Messikh, 2018).

1.2 STATEMENT OF THE PROBLEM

Due to the high efficiency of quantum algorithms and the predicted speed of quantum computers, many big companies such as IBM, Microsoft, and Google- start investing a significant amount of money in these kinds of research to bring quantum computers into reality. Building quantum computers is tough work because the quantum states are susceptible to the external environment and it is difficult to manipulate them without getting a specific error rate. Therefore, there is an earnest request to solve the problem of the external environment to construct robust quantum gates that can be used as a basic unit of the hardware of quantum computers.

Many approaches have been proposed to build quantum circuits. The most successful one is the quantum adiabatic evolution process. This approach is based on the slow change of the Hamiltonian of the quantum states respected to the characteristic time of the system. X. Lacour used this approach to construct the Hadamard single gate and the CNOT and SWAP two-qubit gates using fixed atoms in an optical cavity model. Moreover, L. Giannelli and E. Arimondo extended Lacour’s work and made the process of implementing gates much faster by using a

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superadiabatic approach. The authors introduced an additional Hamiltonian to the main one using the stimulated Raman adiabatic passage with different laser pulses schemes (Giannelli & Arimondo, 2014). Later, Y. H. Issoufa and A. Messikh, investigated the effect of the external environment on the performance of the quantum rotation gate proposed by Lacour and they showed that in an external environment or an open system, where the dephasing occurs in the ground state, the superposition of the state would be broken during the evolution. Moreover, they investigated the Giannelli and Arimondo’s work and discovered that the dephasing reduces the performance of the population transfer, and the fidelity can be far below the quantum computation target even for small dephasing rates (Issoufa & Messikh, 2014).

In 2015, Itay Hen introduced a new approach based on the quantum adiabatic evolutions and gate model. In his approach, he considered the adiabatic evolution of several systems in parallel. This approach allowed him to construct adiabatic single- qubit rotation gates and controlled two-qubits rotation gates. In the same year, Alan C. Santos and Marcelo S. Sarandy discussed the implementation of fast rotation gates and arbitrary n-qubit controlled gates using the controlled superadiabatic evolutions as an alternative to the controlled adiabatic evolutions approach (Santos & Sarandy, 2015).

Itay Hen, Alan C. Santos and Marcelo S. Sarandy implemented their models in a closed system. The closed system is an ideal system, and it does not simulate reality. Indeed, the quantum system has to interact with its surroundings which makes it susceptible to different dephasing effects. Therefore, in this research study, we aim to expand the work done by Itay and make it more realistic, besides, we are going to

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propose a counterpart model to Itay’s model call the controlled nonadiabatic evolutions. The new model can help to accelerate the process of rotating states.

1.3 RESEARCH QUESTIONS

The following research questions are formulated to set the direction of this research.

1. How to implement a universal set of quantum gates using the controlled adiabatic evolutions approach?

2. How to prepare the four Bell’s states using the controlled adiabatic evolutions approach?

3. What is the effect of the environment on the controlled adiabatic evolutions process?

4. What is the alternative approach to the controlled adiabatic evolutions model to accelerate the implementation process of the quantum gates?

5. What is the effect of the environment on the controlled nonadiabatic evolutions model?

1.4 RESEARCH OBJECTIVES

The study aimed to achieve the following objectives:

1. To demonstrate that the controlled adiabatic evolution can be used to generate a universal set of quantum gates.

2. To demonstrate that the controlled adiabatic evolution can be used to prepare the four Bell’s states.

3. Study the robustness of single qubit rotation gates and Bell’s states based on the controlled adiabatic evolutions to the different type of quantum noise.

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4. Propose a counterpart model to the controlled adiabatic evolution that accelerates the evolution process.

5. Study the robustness of single qubit rotation gates based on the controlled nonadiabatic evolutions in an external environment.

1.5 SIGNIFICANCE OF THE STUDY

This study demonstrates that the controlled adiabatic evolutions can be used to implement a universal set of quantum gates that may lead to building some quantum circuits and quantum blocks. Further, this study illustrates how to prepare the four of Bell’s states that can help to implement some communication and security protocols.

Also, during the study we checked the influence of noise on the controlled adiabatic evolutions model to test its reliability in the case of a real environment. We end by Proposing an alternative approach to the controlled adiabatic evolutions that helps to accelerate the rotation process of the quantum gates.

1.6 ORGANIZATION OF THE THESIS This thesis is organized as follow:

Chapter 1 introduces and discusses the problem statement, the objective and the significance of this research.

Chapter 2 is a background chapter that presents the fundamental concepts of quantum computing. We started the chapter by giving a quick introduction about the history of quantum computing and the axioms of quantum mechanics. Next, we introduced the concept of the qubit and its mathematical and physical representation.

Also, we explained, in brief, the theory of quantum gates and their properties. Some

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applications of quantum computing and previous studies were mentioned at the end of this chapter.

Chapter 3 explains the controlled adiabatic evolutions approach. In this chapter, we provide the detail of implementing each single quantum gate using the controlled adiabatic evolutions. Further, we demonstrate that the controlled adiabatic evolutions model can be used to prepare the four Bell’s states.

Chapter 4 demonstrates the noise influence on the controlled adiabatic evolutions model. We introduced a different type of noise in the process of preparing quantum gates, and we measured the outcome of each single quantum gate at the end of the evolution.

Chapter 5 introduces the controlled nonadiabatic evolutions which can be considered as a counterpart approach to the controlled adiabatic evolutions. This new model can accelerate the evolution process. Also, we investigated the effect of noise on this new model.

Chapter 6 reflects the conclusion and the contribution of the thesis. Besides, perspectives and future works are discussed in this chapter.

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CHAPTER TWO LITERATURE REVIEW

2.1 INTRODUCTION

Richard Feynman is the first scientist who introduced the idea of quantum computers.

Feynman claimed that quantum computers would perform computation much faster than classical computers. He said that nature does computation based on quantum mechanics laws, and nobody can do better than nature. Besides, he mentioned that some physical phenomena could only be simulated in quantum systems (Feynman, 1982).

In 1985, David Deutsch proposed the first computational model for quantum computers similar to the classical Turing Machine. His model was named quantum Turing Machine. Later, Deutsch described the theory of the quantum gate and presented a universal set of quantum gates (Deutsch, 1985). In 1992, David Deutsch and Richard Jozsa introduced the first quantum algorithm that can answer whether a given arbitrary function 𝑓(𝑥) is constant or balanced. Their quantum algorithm showed an exponential less time than any classical algorithm (Deutsch & Jozsa, 1992). One year later, Dan Simon proposed a new quantum algorithm using periodicity to solve the black box problem in polynomial time (Simon, 1997). In 1994, Peter Shor discovered a quantum algorithm that can solve the problem of semi- prime number factorisation in polynomial time (P. W. Shor, 1994). Shor’s quantum algorithm may change the future of the information security because today’s information codes rely on the intractability to factor large integers. In 1996, Lov Grover designed a quantum search algorithm to solve the needle in a haystack

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problem (Grover, 1997). Classically, finding a piece of data in an unstructured database of n elements may require n queries but using Grover’s algorithm, the problem can be settled using just √𝑛 queries. Besides these interesting quantum algorithm, several theoretical quantum protocol and applications have been proposed such as, quantum key distribution, quantum teleportation, quantum communication, quantum coding, quantum game, quantum strategies, quantum fingerprinting, quantum internet, etc (Charles H Bennett & Brassard, 2014; C. H. Bennett et al., 1993; Buhrman, Cleve, Watrous, & De Wolf, 2001; Eisert, Wilkens, & Lewenstein, 1999; Kimble, 2008; Meyer, 1999; Schumacher, 1995). The fundamental theoretical quantum computing work was done in the ninety decades after that several experimental works have started to investigate the power behind the quantum world.

Today, Scientists believe that quantum computers will enable us to solve computation problems that currently take the lifetime of the universe in seconds, hours or days. Quantum computers can be used to combat the world’s most challenging problems, e.g. information security, machine learning, fighting disease and creating drugs. The working mechanism of quantum computers depends on quantum mechanics laws. Quantum mechanics is a subfield of physics that study the world nature at the micro level. To understand the quantum computers concept, we need first to learn the basics of quantum mechanics. The next section gives a short history and presents the axioms of quantum mechanics. We recommend reading chapter 1,2 and 3 of the book title: “Quantum Mechanics by Claude Cohen- Tannoudji” for readers who want to learn the necessary quantum mechanics concepts in the quantum computing field (Cohen-Tannoudji & Diu, 1977).

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10 2.2 QUANTUM MECHANICS

Quantum mechanics studies the micro-level phenomena of our universe such as electrons and photons. The first discussed topic in quantum mechanics was the disputed nature of light as a wave or particle. In the late 1660s, Isaac Newton started the first world experiment to understand the nature of light. He displayed his prism to the sunlight and got as an outcome a beautiful spectrum. At that moment, Newton believed that light was a rain of particles because waves do not move in the straight line. At the beginning of the 19th century, the British scientist Young conducted his famous double slit experiment. Young projected a ray of light on a board that contains two small slits near to each other and observed the light passed through the two slits on a screen positioned with a specific distance behind the board. Young noticed light and dark lines between each other. The only explanation to this phenomenon is the light that goes through the first slit has interfered with the light that passed through the second slit which led Young to conclude that light is a wave nature. The world quantum was introduced by Max Plank when he was interested in studying the light phenomena. Planck claimed that light is a particle and he proposed his quantum hypothesis. After, Albert Einstein argued that light has dual wave-particle characteristics and classical mechanics cannot explain this duality. In 1905, Einstein published his light quantum hypothesis that makes him worthy to receive the Nobel prize sixteen years later. The second discussed topic in quantum mechanics was the movement of the subatomic particles (quantum particles). In fact, the behavior of these quantum particles does not obey the Newton laws of motion. In 1926, Erwin Schrödinger developed a probabilistic differential equation of first order that describes the movement of quantum particles efficiently. Below we show some features of quantum particles:

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1. The complete knowledge of an unknown quantum state is impossible.

2. Performing measurement on quantum particle gives only a partial knowledge on its state.

3. We cannot figure out the trajectory movement of quantum particles. We can only know that the particle started at a point x and measure late its position at a point y.

4. Quantum particles have probabilistic behavior. As an example, if we prepare two quantum particles in similar states and measure them, we most probably get different results.

5. A quantum particle behaves simultaneously like a mass and like a wave.

This property of the particle is known as wave-particle duality.

2.3 QUBITS

The bit is the basic information unit of classical computers. The bit can be either 0 or 1 at an arbitrary instant of time 𝑡. The qubit, we call it also a quantum bit, is the basic information unit for quantum computers. The qubit can be either 0 or 1, besides it can be 0 and 1 simultaneously and we call it superposition state of the qubit.

2.3.1 Bra-ket notation

Dirac introduced a mathematical notation to describe a quantum state called bra-ket notation. His notation is widely used in quantum mechanics and the standard presentation of the qubit in quantum information and quantum computing.

|𝑎⟩: is called ket and is a column vector of two dimensions.

⟨𝑏|: is called bra and is a row vector of two dimensions.

The qubit can be defined using the bra-ket notation as follows:

(24)

12

|𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩ where 𝛼, 𝛽 ∈ ℂ ,

𝛼, 𝛽 are called qubit amplitudes or qubit coefficients and they should restrict the

normalization condition |𝛼|2+ |𝛽|2 = 1.

We can write a qubit as a column vector in the following way:

|𝜓⟩ = (𝛼

𝛽) = 𝛼|0⟩ + 𝛽|1⟩ = 𝛼 (1

0) + 𝛽 (0

1) = (𝛼 𝛽).

2.3.2 The Bloch sphere

The Block sphere helps us to describe the state of an arbitrary qubit geometrically on a sphere in 3-dimensional unit Euclidean space. Any qubit can be formulated mathematically using the usual spherical coordinates 𝜃 and 𝜙.

|𝜓⟩ = cos𝜃

2|0⟩ + 𝑒𝑖𝜙 sin𝜃

2|1⟩, (1)

Using these two spherical coordinates (𝜃, 𝜙), we can illustrate any qubit |𝜓⟩

as a point on the Block sphere.

Kulliyyah of

Rujukan

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