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• CZECH TECHNICAL UNIVERSITY IN PRAGUE

Faculty of Nuclear Sciences and Physical Engineering

Department of Physics

DIPLOMA THESIS

V¥rný p°enos informace s poruchami

Perfect state transfer in the presence of perturbations

Author: Bc. Antonín Hoskovec Supervisor: Prof. Ing. Igor Jex, DrSc. Consultant: Aurél Gábris, Ph.D. Academic year: 2011/2012

• Prohlá²ení

Prohla²uji, ºe jsem svou diplomovou práci vypracoval samostatn¥ a pouºil jsem pouze podklady (literaturu, projekty, SW atd.) uvedené v p°iloºeném seznamu.

Nemám závaºný d·vod proti uºití tohoto ²kolního díla ve smyslu 60 Zákona £.121/2000 Sb., o právu autorském, o právech souvisejících s právem autorským a o zm¥n¥ n¥kterých zákon· (autorský zákon).

Declaration

I declare that I wrote my diploma thesis independently, and that I have not made use of any aid (literature, projects, SW etc.) other than those acknowledged.

I agree with the usage of this thesis in the sense of the 60 Act 121/2000 (Copyright Act of Czech Republic).

V Praze dne 4. kv¥tna 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bc. Antonín Hoskovec

• Acknowledgments

I would like to gratefully acknowledge the support from Prof. Ing. Igor Jex, DrSc. and Aurél Gábris, Ph.D. over the past years, the kind understanding for silly mistakes I made, the never ending opportunities for consultations and the immense patience. I truly feel lucky that I was ever able to work with them on the topics investigated, an opportunity not available to many. It was them who introduced me into the fundamental research of quantum computation, they supervised my bachelor's thesis too. I also wish to express my gratitude towards Dr. Georgios M. Nikolopoulos, whom I worked together with on some of the research presented here (namely part IV).

• Název práce: V¥rný p°enos informace s poruchami

Autor: Bc. Antonín Hoskovec

Obor: Matematické inºenýrství / Matematická fyzika

Druh práce: Diplomová práce

Vedoucí práce: Prof. Ing. Igor Jex, DrSc., Katedra fyziky, Fakulta jaderná a fyzikáln¥ inºenýrská, eské vysoké u£ení technické v Praze

Konzultant: Aurél Gábris, Ph.D., Katedra fyziky, Fakulta jaderná a fyzikáln¥ inºenýrská, eské vysoké u£ení technické v Praze

Abstrakt: Teoretický úvod obsahuje základní p°ehled kvantových výpo£t·, od de�nování základních po- ºadavk·, p°es metody na jejich spln¥ní aº po praktické návrhy na jejich uskute£n¥ní. Vlastní výzkum se pak týká p°enosu informace, kdy jsou nejprve diskutovány moºnosti uplatn¥ní známých metod na p°enos informace na uv¥zn¥ných iontech a pozd¥ji jsou v detailu rozebrány poruchy zp·sobené ohyby lineárních °etízk· qubit·. Ohyby jsou o£ekávatelné poruchy p°i manipulaci s informací ve více dimen- zích. V záv¥ru je pak popsána navrhovaná metoda kompenzace za ztráty zp·sobené ohyby. Poruchy byly prozkoumány numerickými metodami.

Klí£ová slova: kvantové po£íta£e, kvantová informace, p°enos stavu, kvantové sít¥ s poruchami.

Title: Perfect state transfer in the presence of perturbations

Author: Bc. Antonín Hoskovec

Abstract: The theoretical introduction part contains basic overview of quantum computation, from de�ning the fundamental criteria, over the methods of ful�lling them, to practical designs of quan- tum computers. The original research is about the state transfer, where �rst application of known methods for state transfer on trapped ions is discussed and then follows the detailed investigation of perturbations caused by bendings of linear qubit chains. The bendings are expected defects when manipulating with the information in more than one dimension. The �nal sections are devoted to description of proposed method of compensation for the losses from bendings. The perturbations were treated numerically.

Key words: quantum computation, quantum information, state transfer, perturbed quantum networks.

• Contents

I Basics of quantum computation and information 8

1 Introduction 8

1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Quantum computation 10

2.1 Quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Universality of Hadamard, C−NOT and π8 gates . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Quantum circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

II Quantum computer candidates 22

3 Quantum harmonic oscillator 22

3.1 Description of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Application to quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Electromagnetic cavities 25

4.1 Description of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 Application to quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Trapped ions 32

5.1 Description of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.3 Application to quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Quantum dots 39

6.1 Description of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3 Application to quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

III State transfer 42

7 Perfect state transfer 43

7.1 Transfer of single excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 Transfer of multiple excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8 State transfer on trapped ions 48

8.1 Illuminating single ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8.2 Illuminating multiple ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

IV Quantum networks with bending 51

9 Investigated protocols 51

9.1 Protocol 1: Uniform couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 9.2 Protocol 2: Optimal couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6

• 10 Analysis of bending loses 54

10.1 Robustness of protocols vs. bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 10.2 Optimization of the transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

V Conclusions 59

References 60

7

• Part I

Basics of quantum computation and

information

1 Introduction

Quantum computation is a term for many growing branches of science, born in 1982 in publications of Richard Feynman [1�3]. The main motivation why quantum computers are so thoroughly investigated, both theoretically and experimentally, is the impact building a quantum computer is expected to have. Impact on perhaps every �eld of science. In 1994 Peter Shor discovered the famous Shor's algorithm for factorization of integers in polynomial time on quantum computers. It is exponentially more e�cient algorithm then the best known classical algorithm - the general number �eld sieve. Shor's protocol is a beautiful example of e�cient quantum computation that would forever change cryptography. The possible bene�ts do not stop here. Let us mention the quantum encryption algorithms, originating 1984 in articles by Charles H. Bennett and Gilles Brassard [4]. More to the point we should call these the quantum key distribution algorithms for secure production and sharing of a secret key that can be used for encryption. From the nature of quantum mechanics it is impossible to eavesdrop such a communication, the communicating parties should be able to detect it. These algorithms are being brought to real life experiments in the present. Quantum key distribution has been presented to work at 1Mbit/s over the distance of 20km of optical �ber. And even much longer distances up to approximately 150km have been achieved by some research organizations. The progress in this �eld went as far as to commercial companies getting involved in the development.

In 1996 Seth Lloyd has shown that any quantum computer could be used for e�cient simulation of any quantum system, even for example DNA. The methods used nowadays for decrypting DNA sequences and their e�ects require large computational power and many times rely on databases of known sequences for comparison of the chemical constitution [5]. The quantum simulations would uncover the e�ects much faster. The simulations might help with further miniaturization of micropro- cessors. Today's companies are capable of producing microprocessors fabricated with characteristic dimensions as low as 45nm. On this level of manipulation with matter quantum behavior is usually considered disturbing, because it allows elec

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