QUANTUM COMPUTING WITH NUCLEAR SPINS

IN SEMICONDUCTORS

a dissertation

submitted to the department of applied physics

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

Thaddeus D. Ladd

June 2005

c Copyright by Thaddeus D. Ladd 2005All Rights Reserved

ii

I certify that I have read this dissertation and that, in my opinion,

it is fully adequate in scope and quality as a dissertation for the

degree of Doctor of Philosophy.

Yoshihisa Yamamoto(Principal Adviser)

I certify that I have read this dissertation and that, in my opinion,

it is fully adequate in scope and quality as a dissertation for the

degree of Doctor of Philosophy.

Alexander Fetter

I certify that I have read this dissertation and that, in my opinion,

it is fully adequate in scope and quality as a dissertation for the

degree of Doctor of Philosophy.

Mark Kasevich

Approved for the University Committee on Graduate Studies.

iii

iv

Abstract

The successful implementation of a scalable, fault-tolerant quantum computer would introduce a type

of information processing more powerful than any available today. Reciprocally, the discovery of a

fundamental obstacle to such a system would be an important advance in the foundations of quantum

theory. No such fundamental obstacles are currently known, but neither has any architecture been

shown to be experimentally scalable.

Many technologies have been considered for finding such an architecture; in this work I focus on

nuclear spins in semiconductors. Semiconductors provide promising optical means for polarizing and

measuring small nuclear spin ensembles, which are tasks that pose critical challenges to quantum

computers based on nuclear magnetic resonance (nmr). At the same time, semiconductor nuclei

are sufficiently coherent quantum oscillators to allow complex information processing using resonant

radio-frequency pulse sequences. In particular, the isotopically clean and magnetically quiet environ-

ment of pure, high quality, bulk single-crystal silicon provides a nuclear environment allowing what

may be the longest absolute coherence time of any solid-state qubit currently under consideration.

I have experimentally tested this claim using high-power nmr pulse sequences to eliminate inhomo-

geneous dephasing and dipolar evolution among an ensemble of 29Si nuclei in isotopically modified

silicon crystals. Intrinsic decoherence processes are only observed in polycrystalline silicon, where

1/f charging noise processes are likely to blame. In high-quality single crystal samples, nuclear

coherence persists for over 25 seconds, a timescale limited only by pulse sequence imperfections.

I will discuss an architecture that takes advantage of this clean nuclear environment, but I will

also address its scalability limitations due to silicons poor optical characteristics. These limitations

will suggest new experiments employing nuclear spins in optically controlled semiconductor quantum

dots, which may hold more promise for future scalable quantum computer architectures.

v

Acknowledgements

This is a dissertation and not a research paper. Were it the latter, the text would be shorter, of

course, but the list of authors would be much, much longer. Many minds contributed to the material

I will present here, and many helped with support and encouragement. I would like to thank some

of them now.

One could not ask for a better adviser than Yoshi Yamamoto. His research program is expansive

and dynamic, and yet he is able to be a very supportive manager to all of his students. And although

it took me some time to fully realize it, he is above all else a great educator. He has guided me

toward successes but has also let me make mistakes; I understand now that both were required to

allow the kind of professional growth through which he has guided me. His approach to solving

problems reminds us that great challenges are great challenges and small issues are small issues, and

that throughout all technical problems, basic physics and engineering principles usually provide the

answers. His clear manner of thinking, his vision, and his ambition have all been inspirational.

Maintaining clear thinking, vision, and ambition in the darkness of the lab can sometimes be

exhausting, and here the other students and researchers of the Yamamoto group have been guiding

lights for me. I must first thank the resourceful Jonathan Goldman, with whom I have worked

most closely in these seven years. Jonathan and I arrived to the Yamamoto group together and

undertook the daunting task of building a new lab for nmr, micromagnetics, magnetic resonance

force microscopy, and magneto-optics, starting with no equipment, no labspace, and no technical

expertise. For the expertise, we both spent an educational summer in Japan. In my case, Atsushi

Goto, Tadashi Shimizu, and others at the National Research Institute of Metals in Tsukuba, were

extremely helpful for teaching me the basics of low-temperature solid-state nmr. Recently, Jonathan

and I have built yet another new laboratory from scratch, something no graduate student should

have to do twice. Jonathans hard work deserves more recognition that it has so far received.

Many of the actual experiments described in this dissertation were principally aided by the

industrious Denys Maryenko. Denys arrived as an undergraduate student at the same time as

vi

another helpful individual worthy of acknowledgement, Ernest Yeung. Motivated by their own

ambition, these two kicked off the new nmr lab with early relaxation experiments in insulating

crystals, including fluorapatite samples grown by Professor Ian Fisher, who was extremely gracious

to offer so much time and expertise to our project. Denys returned for his Masters thesis and

began the experiment to measure long decoherence times in silicon. He did much of the work in

this experiment, including the construction of the nmr probe that made the results possible and

the programming of the pulse sequences. More importantly, his enthusiasm for the work pushed it

forward in times of difficulty.

Although few in the Yamamoto group have worked directly on nmr, everyone in the group has

taught me something. In particular, Will Oliver taught me much about high frequency electronics,

and Na Young Kim has also been a great help in this arena. Charlie Santoris expertise in single-

quantum-dot optical measurements has been a great resource; he is a true scientist, a die-hard

skeptic capable of bringing out the optimist in the rest of us. Since Charlie left the group, Stephan

Gotzinger has nicely filled his shoes. Kai-Mei Fu is a dynamo in the lab, bringing energy and ideas

to the most challenging of experiments. When Anne Verhulst joined the lab, she brought useful nmr

knowledge that was quite valuable. I must acknowledge Edo Waks for always having the answer to

any question I ask him, and always taking the time to provide it. Most recently Andrei Faraon has

started his graduate career partly in our laboratory, where his help has been much appreciated. Our

recent work with quantum dots would also not be possible without the growth expertise of Glenn

Solomon and Bingyang Zhang.

Much of my thesis project, however, has been theoretical, and discussions with group members

in this arena deserve acknowledgement. The whole idea of crystal lattice quantum computation was

initiated by Fumiko Yamaguchi, whose open-minded ideas have seeded a large portion of my work.

Mike Jura and Matt Terrel were visitors to our group early in their careers where they helped me

work on the theory of optical polarization of nuclei in semiconductors. And Cyrus Master has always

been a brilliant theoretical consultant.

I have benefited from all the other members of the Yamamoto group as well, whether from

discussions in the hallway, inspirational group-meeting talks, pointed questions, or just good times.

So I extend thanks to Oliver Benson, Aykutlu Dana, Hui Deng, Leo Di Carlo, Eleni Diamanti,

David Fattal, Dehuan Huang, Robin Huang, Kyo Inoue, Jungsang Kim, Shinichi Koseki, Prab

Kuniyil, Debbie Leung, Xavier Matre, Matt Pelton, Jocelyn Plant, David Press, Patrik Recher,

Kaoru Sanaka, Barry Sanders, Masa Shirane, Mitsuro Sugita, Jelena Vuckovic, and Gregor Weihs.

The administrative help in the Yamamoto group is also world-class, thanks to the hard work of

vii

Yurika Peterman, Mayumi Hakkaku, and Rieko Sasaki.

Much of this research, however, has benefited from collaborations outside of Stanford. In partic-

ular, my collaboration with the group of Kohei Itoh at Keio University in Japan has been extremely

fruitful. Kohei and his students Eisuke Abe and Rodney Van Meter have helped me not only by

providing isotopically engineered silicon, but also with their ideas and their extraordinary vision.

Naveen Khaneja at Harvard has been a very helpful resource; he suggested mrev-16 as a way to

remove effective fields in decoupling experiments, and as I will discuss this suggestion turned out

to be more helpful than either of us anticipated. At Berkeley, Anant Paravastu and Pat Coles in

the group of Jeff Reimer have taught me much about optical polarization in semiconductors, and I

am impressed by their intelligence and patience in the face of the challenge of understanding their

data. Several people at the ibm Almaden Research Center were helpful for various reasons: Ike

Chuang, for both his long-term vision and his practical advice; Nino Yannoni for sharing his infi-

nite reservoir of nmr expertise; and Bruce Gurneys research group, including Jeff Childress and

Matt Carey, for providing me an enjoyable summer internship. Finally, the group of David Cory at

mit, especially Chandrasekhar Ramanathan, Jonathan Baugh, and HyungJoon Cho, have recently

brought my understanding of nmr to the next level.

I must thank the many, many people who wrote the code for LATEX2e and provided it for free.

Without this typesetting software the development of a document of this complexity and length

would have been much, much harder.

I would also like to thank the members of my oral defense and reading committees, not only for

patiently hearing and reading about my ideas but for their own inspirational work. Mark Kasevich

taught me much in my three quarters as his teaching assistant, and his research program is inspira-

tional. Sandy Fetters class in electromagnetism was the best I took at Stanford, and perhaps some

element of the style of his notes has carried on into this dissertation. Walter Harrison has written

nearly all of the papers and books in solid-state theory that actually increased my comprehension of

the subject. I also thank my chair, Jody Puglisi, for bringing his nmr expertise to the committee.

I also owe a huge debt of gratitude to the Fannie and John Hertz Foundation. Their monetary

support was extremely generous, and the discussions with directors (Lowell Wood in particular)

and other fellows at interviews and symposia helped me maintain my direction, especially in the

beginning.

I must also acknowledge Henry Chin, who has provided kind friendship amongst much advice,

both technical and fuzzy. And finally, my extremely brilliant wife, Sharon Ungersma, for her love,

intelligence, guidance, support, and encouragement that, ultimately, make all of this worthwhile.

viii

Contents

Abstract v

Acknowledgements vi

1 Introduction 1

1.1 Is Quantum Mechanics Complete? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Einstein and Local Realism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Schrodinger and Verschrankung . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Theoretical Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 The Deutsch-Josza Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Shors Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Other Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.4 Quantum Error Correction and Fault Tolerance . . . . . . . . . . . . . . . . . 12

1.3 Experimental Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Physical Resources for Quantum Computing 21

2.1 Three Essential Requirements for Fault Tolerant Quantum Computation . . . . . . . 21

2.1.1 Scalable Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.2 Universal Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.3 Initialization/Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Opening the Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Quantum Memory and Quantum Repeaters . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 The Reasons for Quantum Communication . . . . . . . . . . . . . . . . . . . 34

ix

2.3.2 The Quantum Repeater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.3 Physical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Quantum Computing Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.1 Atomic and Molecular Optics Implementations . . . . . . . . . . . . . . . . . 41

2.4.2 Solid-state Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4.3 Linear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 NMR Quantum Computers 55

3.1 NMR Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Liquids vs. Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.2 Distinguishing Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.3 Quantum Logic and Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.2.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2.5 Are Liquid State Quantum Computers Really Quantum? . . . . . . . . . . . 82

3.3 Theory of Indirect Nuclear Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3.1 Effective Hamiltonians from Second Order Perturbation Theory . . . . . . . . 84

3.3.2 One Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.3.3 Two Coupled Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.3.4 Lattice of Coupled Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.4 The All-Silicon Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.4.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.4.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.4.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.5 Quantum-Dot NMR Quantum Computers . . . . . . . . . . . . . . . . . . . . . . . . 102

3.5.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.5.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.5.3 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4 Decoupling: Theory 107

4.1 Ensemble vs. Single-Spin Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.2 Theory of Dipolar Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2.1 Multiple Pulse Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2.2 Magic Angle Spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

x

4.3 Spin Echoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3.1 The Effects of -pulse Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3.2 Spin-echo Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.3.3 Even-Odd Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.4 Recoupling Pulse Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5 Decoupling: Experiment 131

5.1 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.1.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.1.2 Samples and Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.1.3 The Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.1.4 Small Angle FID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.1.5 Pulse Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.1.6 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.1.7 Spin Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6 General Relaxation Theory 153

6.1 Fermis Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.2 General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.2.1 System and Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.2.2 Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2.3 Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.2.4 Observable Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.3 Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3.1 Spin-Boson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3.2 General Formulae for T1 and T2 . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.3.3 Two Hyperfine Coupled Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7 Nuclear Relaxation in Silicon 177

7.1 Free Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.1.1 Conduction Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.1.2 Valence Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.2 Trapped Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.2.1 Stationary Impurity Donors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

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7.2.2 Optically Excited Bound Excitons . . . . . . . . . . . . . . . . . . . . . . . . 186

7.3 1/f Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.3.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.3.2 Decoherence Due to 1/f Noise During Decoupling . . . . . . . . . . . . . . . 190

8 Prospectus 197

A Notation 199

A.1 Vectors, Matrices, and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

A.2 Quantum Mechanics and NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

A.3 Liouville Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

A.4 Many-body notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

A.5 The Hyperfine Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

B Spin Algebra 207

B.1 Single Spin Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

B.2 Two Spin Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

B.3 Diagonalizing a 2 2 Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 210

Bibliography 211

Index 229

xii

List of Figures

1.1 A comparison of execution times of classical and quantum factoring algorithms vs.

the size L, in bits, of the number to be factored. The curve labelled nfs represents

the expected scaling of the classical nfs algorithm with the total scale set by the

current world-record largest implementation of it, in which 104 personal computers

running in parallel factored a 576-bit number in one month in 2003, according to

rsa Security, Inc. The curves labelled Shor C represent an implementation of Shors

quantum algorithm in which logical qubits may only couple to nearest neighbors. The

curves labelled Shor E represent an implementation in which any logical qubit may

couple to any other logical qubit. These implementations are described by Van Meter

and Itoh [13]. Both algorithms use 100L logical qubits. For comparison, clock speeds

of 1 Hz or 1 MHz are shown; these clock speeds must include the time for extra

processing for quantum error correction. . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 A quantum circuit representing the 3-qubit bit-flip code. The qubit represented by the

top line is encoded into three qubits with two controlled-not operations on two ancilla

qubits. A single qubit error occurs somewhere in the dashed box. The syndrome and

recovery operations are implemented with two controlled-not operations and two

polarization measurements; the top qubit is flipped if both ancilla are measured in

state |1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 The number of pure qubits n which may be extracted at finite polarization as a

function of the polarization p. Each curve is labelled by the total sensitivity, which is

the number of qubits one is capable of measuring divided by the number of qubits in

the ensemble (or the number of repetitions of the experiment). . . . . . . . . . . . . 29

xiii

2.2 A modification of the quantum error correction circuit shown in Fig. 1.2, in which the

measurements with feed-forward control have been replaced by a coherent Toffoli gate

for recovery. After the recovery, the ancilla qubits must be discarded or reinitialized. 31

2.3 Schematic of an optical repeater. Optical beams a and b meet at a beam splitter,

and the output beams are sent to distant qubits A and B. Each qubit is an identical

-system, as shown in the center of the figure. The two qubit states are the two

lower states of the system; the excited state is optically accessible only to state |0.Many photons are lost at A and B, but some photons reflect off A or B if they are in

state |0. These reflected photons are combined at a second beam-splitter, completinga Mach-Zender interferometer; a detection event at port c may project the distant

qubits into an entangled state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1 The dipolar magnetic field BD(r) generated by one nucleus is seen by a random

assortment of other nuclei. Only the z-component of the dipole field is important at

high applied fields B0; this is the secular component. This component vanishes at the

magic angle M shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Sample geometries for generating a large magnetic field gradient across a regular

crystalline lattice. The magnetic field in the macromagnet causes planes of nuclei to

have unique Larmor frequencies j . Calculations of expected magnetic field gradients

for these geometries appear in Ref. 141. . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3 The basic geometry for a two-dimensional magnetic field gradient. The gradient field

G points at a slight angle from the x-axis. The gradient causes each nucleus in the

grid to have a unique Larmor frequency j . . . . . . . . . . . . . . . . . . . . . . . 75

3.4 An illustration of the geometry of nuclei discussed in the text. The nuclei form a

periodic lattice. The applied magnetic field and magnetic gradient are assumed to

point in the z-direction, so horizontal rows of nuclei have the same Larmor frequency

j . The nearest-neighbor qubit-qubit distance is a and the nearest-neighbor distance

among ensembles is 1a. If the angle is the magic angle, then 1 =2. . . . . . . 78

3.5 A schematic for a fluorapatite crystal lattice quantum computer. The fluorapatite

crystal is shown mounted on a silicon cantilever, whose oscillations provide mrfm

readout. The cantilever and crystal are aligned with the micromagnet which generates

the field gradient. The inset shows the one-dimensional structure of 19F nuclei in

fluorapatite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xiv

3.6 A schematic for an all-silicon quantum computer. The figure shows the integrated

micromagnet and bridge structure needed for distinguishing qubits and for read-out.

The bridge has length l = 300 m, width w = 4 m, and thickness t = 0.25 m. The

micromagnet has D = 400 m, L = 4 m, and W = 10 m, and produces a field

gradient of Bz/z = 1.4 T/m, uniform over a 100 m by 0.2 m region inside the

bridge. The insert shows the structure of the silicon matrix and the terrace edge. The

darkened spheres represent the 29Si nuclei, which preferentially bind at the edge of

the Si step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.7 Images of single atom chain growth in silicon. A stm image appears on the left with

corresponding ball-and-stick atomic model on the right; the arrows on the bottom

compare critical atomic locations between the two images. The images labelled (a)

show a terrace-step on the Si(111)77 surface before deposition of additional silicon;this structure would be made from spin-0 28Si. The images labelled (b) show that

deposited silicon atoms form a straight atomic chain on the terrace step; these extra

atoms would be the spin-1/2 29Si nuclei. This figure is taken from Ref. 171, which

explains the experimental conditions and the modelling in more detail. . . . . . . . 95

3.8 Energy diagram for the neutral donor (P0) and its bound exciton (P0,X) in a magnetic

field. The (P0,X) state is populated via capture of a resonantly excited free exciton.

The 31P nuclear state can be determined by the energy difference of a and b. . . . . 99

4.1 Simulated dipolar decoherence versus time. The broken lines represent individual spin

measurements Re{Gj(t)} on an 8-spin simulation of dipolar evolution with uniformlyrandom coupling constants with range [D0/2, D0/2]. The solid line is the magnitude-sum |M(t)|. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2 The wahuha pulse sequence. All pulses are broadband /2 pulses of the indicated

phase. Measurements are made at the times marked S; here the toggling and rotating

reference frames coincide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.3 Schematic of the cpmg-mrev-16120 pulse sequence with spin-echo data. The echoesshown in the upper left and expanded in the upper right are data from an isotopically

natural single crystal of silicon. These are obtained by first exciting the sample with

a /2-pulse of arbitrary phase , decoupling with the mrev-16 sequence shown in

detail on the bottom line, and refocusing with -pulses of phase = + /2 every

120 cycles of mrev-16. The magnetization is sampled once per mrev-16 cycle in the

windows marked with an S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xv

4.4 A sample pulse sequence incorporating Hadamard decoupling and wahuha. This is

constructed directly from H(4) by translating each change of sign in the ith row to

a soft RX() pulse, labeled , at frequency i. The first, all positive, row of H(4)

has been removed to eliminate inhomogeneous broadening. Single qubit rotations, as

indicated by , must occur between full cycles of the sequence. Recoupling between

qubits i+ 1 and i+ 2 may be achieved by inserting a pulse where indicated by the

dotted line. The top row represents a broadband wahuha sequence for homonuclear

decoupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.1 The series parallel tank circuit with an extra Q-reducing capacitor C. . . . . . . . . 135

5.2 A typical data time series and Fourier transform showing clear spin locking effects.

This data was taken with polycrystalline silicon under mrev-8. The raw time series

on the left shows the first second of the real (in-phase) and imaginary (out-of phase)

amplitudes; the waterfall plot on the right shows the result of applying the dft to

each echo block as described in Sec. 5.1.6. . . . . . . . . . . . . . . . . . . . . . . . . 144

5.3 The magnitude of the center and side peaks from a waterfall plot when -pulses

are applied frequently (N = 5) without the cpmg convention under mrev-8 with

tc = 1.03 ms, again taken with polycrystalline silicon. These correlated oscillations

in the magnitude are due to -pulse errors. . . . . . . . . . . . . . . . . . . . . . . . 146

5.4 Oscillations observed in the imaginary part of the waterfall side peak when using

alternating -pulse phases with an orthogonal preparation pulse, except where labelled

cpmg where the pulses are of constant phase. The oscillation frequencies vary with

the sparsity of the -pulses (N) as shown in the plot on the left and with the frequency

offset as shown in the plot on the right, as expected for -pulse errors. The oscillations

are removed by using the cpmg convention. This data was taken with polycrystalline

silicon under mrev-8 with tc = 0.681 ms. . . . . . . . . . . . . . . . . . . . . . . . . 147

5.5 Pulsed spin locking observed in heavily doped silicon wafers under cpmg-mrev-16Nwith tc=2.46 ms. The magnitudes of the side-peak echoes are shown. . . . . . . . . . 148

5.6 The cpmg sequence in heavily doped silicon wafers without decoupling. The different

traces correspond to different -pulse spacings 2 . Long-lived echoes are likely due

to spin-locking effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xvi

5.7 Coherence time vs. cycle time in single-crystal silicon. The solid line is a fit showing

the exponent 2.09 0.07 for the isotopically enhanced sample (left) and 2.00 0.2 for the isotopically natural sample (right). The insets show the integrated log-

magnitude of the spin-echoes decaying in time for a few cycle times. . . . . . . . . . 149

5.8 The decay of the spin-echo peaks under mas for several rotation speeds , with

exponential fits (left), and the observed decay times T2 plotted against (right). . 150

5.9 Spin echoes for isotopically depleted silicon under cpmg-mrev-16120. The solidline shows exp(t/8 sec), for comparison. . . . . . . . . . . . . . . . . . . . . . . . . 151

5.10 Echo decay curves for pure polycrystalline silicon of natural isotopic abundance. No

significant variation in the data is observed as tc is changed. The solid line is a fit to

the function described in Sec. 7.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.1 A schematic of the magnetic noise spectral density 2J() in silicon. (The direction

of the magnetic noise vector is neglected in this plot). The Lorentzian magnetic noise

spectral density due to a very fast fluctuator (white noise) is represented by the dashed

line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

7.2 Residuals versus time. The deviation of the data of Fig. 5.10 from the fitting function

of Eq. (7.48), with a histogram of those residuals on the right, consistent with Gaussian

noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

xvii

xviii

Chapter 1

Introduction

The more success the quantum theory has, the sillier it looks.

Albert Einstein

I dont like it, and Im sorry I ever had anything to do with it.

Erwin Schrodinger

1.1 Is Quantum Mechanics Complete?

The theory of quantum mechanics tells us that the world is much, much more complex than what

we observe. It posits the bizarre notion that the results of physical measurements do not exist

until the actual measurements are made. In the meantime, our universe is described by its wave

function, a very large mathematical description allowing the universe to simultaneously coexist in

contradictory configurations.

The superb agreement between quantum electrodynamic calculations and experimental measure-

ments of simple systems such as energy levels of the hydrogen atom might suggest that, insofar as

energies are low and only electromagnetic forces are at play, this theory is final, a closed book.

Frontiers in the quest for a more complete theory are then pushed to the other forces, to extremely

high energies, and to gravity, where general relativity must somehow be fused with quantum field

theory. Physics does not yet have a complete theory to explain the processes during the big bang

or in a black hole, and those who seek to work at the cutting edge of theory often find themselves

in these cramped, uncomfortable times and places.

2 Chapter 1. Introduction

But if we go as far as believing that the predictive power of the wave function indicates that

it represents physical reality, then there must be another frontier in physical theory: finding the

edge between the quantum world and the classical world. How can we reconcile the multifarious

possibilities of the wave function, verified in observations of microscopic quantum systems, with the

much simpler observations of the macroscopic world?

1.1.1 Einstein and Local Realism

One approach to this question is to suggest that quantum theory is simply incomplete. This approach

was taken by Einstein, whose famous 1905 paper on the photoelectric effect and support of the

ideas of de Broglie played critical roles in the establishment of quantum theory. In 1935, Einstein,

with coworkers Podolsky and Rosen [1], argued that quantum mechanics must be missing hidden

variables. The route to contradiction was to show that the act of quantum measurement would

violate the principles of relativity. The resulting paradox is known as the epr paradox.

The act of measuring a quantum system inevitably changes that system, an interpretation held

from the early beginnings of quantum theory in Copenhagen. Before a measurement, we have the

wave function, a mathematical representation of the quantum state, but prior values of particular

measurements are not contained in this wave function in any deterministic sense. For each mea-

surement, these values are randomly selected with a probability distribution that may be calculated

given the wave function. After the measurement, the wave function collapses into the measured

state, and as the theory is written this collapse happens instantaneously over the entire wave

function, even if the wave function describes spatially distant, non-interacting objects. In this sense,

quantum theory is notlocal. If wave function collapse is a physical action, Einstein argued, it

allows instantaneous action at a distance, a notion prohibited by the tenets of relativity. The act

of quantum measurement was therefore contradictory to Einstein, and he argued that additions to

the theory would be required to save it. Einstein preferred a theory in which the results of any

measurement were physically present before the measurement. The apparent randomness in this

theory would be a result of hidden variables currently unknown to the theory. Such a theory would

be realistic or objective.

This line of thinking developed substantially after Einsteins time. The crucial step came in 1964

when Bell [2] developed quantitative inequalities to differentiate the statistics of measurements of a

system described by a strictly local, realistic theory from the measurements predicted by standard

quantum mechanics. In the subsequent decades, multiple experiments violating Bells inequalities

1.1 Is Quantum Mechanics Complete? 3

showed, shy of obscure loopholes, that quantum mechanics is indeed either non-local or not realis-

tic. Einsteins famous intuition failed him in this case; the modern researcher brands Einstein, quite

ironically, as classical, and insists on a new, more open-minded intuition. Each of us who accepts

quantum mechanics must choose whether we prefer to abandon either locality or realism, for

quantum mechanics cannot have both.

1.1.2 Schrodinger and Verschrankung

Schrodinger might be considered the inventor of the wave function, although to him the wave function

represented a real, physical wave, not a generalized probability distribution. This latter interpreta-

tion is due to Born, Heisenberg, and especially Bohr. Schrodinger sided with Einstein in finding fault

with the Copenhagen interpretation. He attempted to bring the Physical Review paper of Einstein

et al. [1] to popular attention with an article in the German magazine Naturwissenschaften (Natu-

ral Sciences) [3]. In this article, Schrodinger identified the heart of the epr paradox, the existence

of states featuring multiple quanta with a non-classical kind of non-local correlation. He called this

concept Verschrankung, an ordered intertwining, which has been translated as entanglement. To

popularize the strangeness of entanglement, Schrodinger asked the reader to imagine a macroscopic

entangled object. His choice was a cat, brought to an entangled state in which, due to a carefully

contrived series of events beginning with a single nucleus in a superposition state, the cat persists

in a state simultaneously alive and dead. This was supposed to be so absurd that, like Einstein,

the reader would assume something was wrong, or at least incomplete, about quantum mechanics.

Schrodingers famous misfortuned cat challenges a naive notion about quantum mechanics: that it

applies only to physical systems whose action is of order ~. In fact this is not built into the theory,

and there is no fundamental limit in quantum mechanics to the size of entangled states, allowing

such bizarre possibilities as Schrodingers cat.

More recently, the concept of decoherence has provided a functional resolution of Schrodingers

dilemma. Decoherence may be summarized as the following process: when information about degrees

of freedom of a large system are ignored or lost, quantum superpositions (including entanglement)

lose phase coherence, causing descriptions of states to revert to classical probability distributions.

The mathematical part of this definition is straightforward, as I will discuss in Sec. 2.2.1 as the partial

trace over a density operator. The premise of it, however, simply moves Schrodingers concerns about

the theory to a different place, rather than eliminating them. When exactly is information lost or

ignored? Disturbingly metaphysical is the following question: Does decoherence require a conscious

observer?

4 Chapter 1. Introduction

Many interpretations of quantum mechanics answer these questions in different ways. Some

theories posit new physical effects that introduce a fundamental decoherence as the size of the

system grows; a particularly tempting line of thought connects this fundamental decoherence to

gravity. A useful review of the state of this line of questioning is given by Leggett [4]. I believe that

these questions cannot be answered strictly theoretically. Currently, the interpretation of quantum

mechanics is a matter of taste. However, an experimental route to further explore these questions

is available, just as measurements of Bells inequalities have answered the epr paradox in favor of

quantum mechanics without local realism. If quantum mechanics is complete, at least on laboratory

length and time scales, then it should be possible to build arbitrarily large entangled states of matter,

such as Schrodingers cat. Introduced as a notion of absurdity in 1935, Schrodingers cat has become

an experimental goal pursued by physicists around the globe.

Certainly, states exhibiting coherent quantum entanglement among many quanta have been

observed in various systems. Obvious examples are bosonic condensates such as superfluid helium

and Bose-Einstein condensates of optically trapped atoms. Coherent states of macroscopic numbers

of fermions are also observable; superconductors and the quantum Hall effect provide examples.

However, these systems exhibit entanglement only for a small subset of states, which are either close

to the energetic ground states of the quantum system or are stabilized by measurement. Closer

analogs to Schrodingers cat have been approached in various experiments; examples include a

superposition of distinct optical coherent states in a microwave cavity [5] and in distinct states of

superconducting Josephson junction circuits, which I discuss further in Sec. 2.4.2. Even in these

experiments, the available number of quantum states and the timescale during which they survive

before extrinsic decoherence processes wash them out is limited. A system to truly satisfy the qualms

of Schrodinger should have access to the entirety of its Hilbert space.

How large is Hilbert space? There is truly a gestalt to Verschrankung, as demonstrated by the

following argument. For this, we enumerate the state space of n quantum two-state systems, known

as quantum bits or qubits. A single qubit is not impressive; it is easily simulated by a single classical

oscillator. The number of states available to a qubit can be enumerated by considering all operations

that can be performed on a qubit in its ground state. This is the group of 22 unitary matrices withunity determinant, SU(2). Physically, this is the group of norm-preserving rotations in 3 dimensions.

It has exactly 3 generators, Iz, Ix, and Iy. These may respectively correspond to 3 measurements

of a classical oscillation: the dc bias, the in-phase amplitude, and the out-of-phase amplitude. In

the quantum case, measurement of the mean of the binomial random variable corresponding to

these three generators would provide enough information to completely reconstruct the state. If we

1.2 Theoretical Quantum Computing 5

neglect all entangled states, the number of states available to n qubits corresponds to the number

of operations in the group [SU(2)]n; the direct product of SU(2) with itself n times. This group

has 3n generators, and is no larger than the space available with n classical oscillators, each with

relative phase, amplitude, and bias. But the postulates of quantum mechanics dictate that Hilbert

space is much, much larger than this. Nothing in the theory prohibits arbitrary unitary operations

on n qubits, so the group of available operations is SU(2n), a group much larger than [SU(2)]n.

The group SU(2n) has 4n 1 generators, which means an exponentially large number of statisticalmeasurements would be needed to collect all the quantum information in an unknown state of n

qubits!

The quantum computer is a system that exploits the inherent non-classicality of a gigantic

Hilbert space; it is a generalized Schrodingers cat. It is a device designed to take a reasonably

large number of quanta and put them into an arbitrary highly entangled state. Then, once these

macroscopic entangled states are available, their existence can be demonstrated without a decohering

measurement, by using them to compute information in ways no classical computer could. If such a

quantum computer is built, it would act as a new variety of Bell inequality measurement, not testing

local realism but rather testing whether quantum mechanics holds up to complexity.

Building very large, persistent entangled states requires the suppression of decoherence, and by

extension the suppression of measurement. This would be extremely impractical if it were not for the

possibility of quantum error correction, a process whereby quantum information is protected from

certain kinds of decoherence. The discovery of algorithms possible on quantum computers with

quantum error correction is a recent and revolutionary development of quantum mechanics. It has

shifted the direction of research in the foundations of quantum theory from Is quantum mechanics

complete? to If quantum mechanics is complete, what can we do with it?

1.2 Theoretical Quantum Computing

The quantum computer did not arise out of the considerations described in the previous section,

but rather out of fundamental considerations in computer science. It was at about the same time

as Einsteins and Schrodingers attacks on quantum theory that Turing [6] introduced what has

become known as the Universal Turing Machine, and suggested that any algorithmic process can be

performed on such a machine, a suggestion that has become known as the Church-Turing thesis. As

digital computers became more and more powerful in the 1960s and 70s, an important modification

to this thesis arose, which is that the efficiency of an algorithm is also important. The strong

6 Chapter 1. Introduction

Church-Turing thesis says essentially that if one physical device can perform an algorithm, then a

Universal Turing Machine can perform the same algorithm without a substantial difference in the

number of computational steps.

One kind of algorithm that was noted by many to be inefficient on any classical computer was the

simulation of large quantum systems. This is because the wave-function has exponentially growing

complexity for a linearly growing number of quanta. In 1982, Feynman [7] turned this situation on

its head by suggesting that the efficient simulation of large quantum systems should be possible with

a quantum mechanical computer [7].

A logical extension to this idea is that a quantum computer may efficiently perform an algorithm

that a Universal Turing Machine cannot, challenging the strong Church-Turing thesis. It was Deutsch

[8] who first sought and found a specific example of such an algorithm, which was subsequently

expanded with Josza into the Deutsch-Josza algorithm [9].

1.2.1 The Deutsch-Josza Algorithm

The Deutsch-Josza algorithm is the simplest of quantum algorithms, and will therefore provide

an illustrative example. Despite its simplicity, it features most of the key elements that a quantum

computer will be expected to perform. It is certainly a quantum algorithm that completes its task in

exponentially fewer steps than its classical analog. However, it presumes the presence of an oracle,

a black-box part of the algorithm whose implementation-time is not considered in the complexity

measure, and which must itself be quantum mechanical. The algorithm must use this oracle in order

to find out what the oracle does.

It is known before the algorithm begins that the oracle takes in a string of n-bits, x, and for

each of the 2n possibilities it returns a single bit corresponding to function f(x). It is promised that

either f is constant, meaning f(x) returns zero for all x or one for all x, or f is balanced, meaning

f(x) returns zero for exactly half of the 2n input strings x and one for the other half. The goal of

the algorithm is to deduce whether the function f is constant or balanced while calling the oracle a

minimum number of times. This artificial problem was invented for the sake of the algorithm; it is

not known to have practical application.

A classical computer cannot deterministically evaluate whether f is constant or balanced without

calling the oracle 2n1+1 times. If after 2n1 different queries f always returns a single value, both

possibilities are still viable; one more call is needed to eliminate the possibility of the constant

function1. A quantum computer, however, assumes that the oracle is linear, so that if you may

1Note, however, that if the a priori probability of each possibility is equal, then it becomes exponentially less

1.2 Theoretical Quantum Computing 7

input one bit-string to it, you may insert all strings of qubits at once.

A linear superposition of all bit strings is obtained by applying the single Hadamard gate, H , to

each bit. This is sometimes referred to as the Walsh-Hadamard gate. This gate takes a single qubit

from |0 to (|0 + |1)/2 or from |1 to (|0 |1)/2, hence converting the polarization of thequbit to the phase of the qubit2. It is easily seen that if the Hadamard gate is applied one-by-one

to a register of qubits, it takes any bit string represented by those qubits to a superposition of all

possible bit strings, with the information about the original string encoded as the relative phases of

the terms in the superposition. For example, if we apply this gate one-by-one to each qubit in the

string |010, we obtain

H3 |010 = H |0 H |1 H |0

=18

[(|0+ |1

)(|0 |1

)(|0+ |1

)]

=18

[|000+ |001 |010 |011+ |100+ |101 |110 |111

].

The generalization to n-qubits may be written

Hn |x = 12n

y

(1)xy |y . (1.1)

The Deutsch-Josza algorithm begins with the register of n qubits all in the |0 state and oneancillary qubit in state |1. To this we apply the (n+ 1)-qubit Hadamard gate:

H(n+1) |0n |1 = 12n+1

x

|x(|0 |1

). (1.2)

We now have our superposition of all possible states which we feed to the oracle Uf . The oracle has

the form of a reversible many-qubit gate, so the initial information must be preserved. It therefore

has the form

Uf |x, y = |x, y f(x) , (1.3)

where indicates bitwise addition (i.e. addition modulo 2). When Uf is applied to the linear

probable after each random call that the function is balanced. In this sense the Deutsch-Josza algorithm shows littleimprovement over a probabilistic classical algorithm.

2To be precise, the polarization of the qubit is p = Iz/I and the phase is = i log[I+/I+III+]. In a

Bloch sphere representation, p is the sine of the latitude and is longitude.

8 Chapter 1. Introduction

superposition of Eq. (1.2), the result is

UfH(n+1) |0n |1 = 1

2n+1

x

(1)f(x) |x(|0 |1

). (1.4)

Here we see Schrodingers Verschrankung. Although only one call has been made to the oracle, the

information about an exponentially large number of calls to f is stored in the phases of a many-

body entangled wave function. For the quantum computer to be universal, any such entangled wave

function must be physically possible.

The algorithm is not yet complete, though, because the relative phases contained in the wave

function are not directly measurable. A polarization measurement made at this stage of the compu-

tation would show the computer in any of its possible states with equal probability! To be useful,

the phases of the wave function must be converted back to polarization, which may again be done

with the (n+ 1)-qubit Hadamard gate:

H(n+1)UfH(n+1) |0n |1 = 1

2n

x

y

(1)xyf(x) |y |1 . (1.5)

Now, the polarization measurement of the register has the following probability of yielding the initial

state |0n |1:

|1| 0|nH(n+1)UfH(n+1) |0n |1|2 = 12n

x

(1)f(x)2

=

1, f is constant

0, f is balanced.

(1.6)

Even though the final measurement is stochastic, the initial promise that f is constant or balanced

assures that if any single qubit of the first n qubits is measured in the state |1, the function must havebeen balanced. If this promise is not made, then the algorithm offers no deterministic information

about the function.

1.2.2 Shors Algorithm

The Deutsch-Josza algorithm provided some idea that quantum computers were powerful. Key to

the algorithm was the initial conversion of the polarization of each qubit to the phase of that qubit,

performed by the Hadamard transformation. This is essentially the idea of the Fourier transform.

The classical Discrete Fourier Transform (dft) takes a vector of N complex values g to a vector of

1.2 Theoretical Quantum Computing 9

N complex values G with components

Gk =1N

N1j=0

gje2ijk/N . (1.7)

The Hadamard transformation is a single-qubit Quantum Fourier Transform (qft). Consider a

qubit in state g0 |0+ g1 |1. The dft of its amplitudes gives

G0 =12(g0 + g1)

G1 =12(g0 g1).

The Hadamard transformation applied to our qubit is G0 |0+G1 |1.

The dft is routinely used to find periodicity in data, and indeed, in 1994 Simon [10] showed an

algorithm using Hadamard transformations for finding the period of a function. The truly dramatic

leap in the field of theoretical quantum computation, however, came when Shor [11] generalized from

the Hadamard transformation to the n-qubit qft, and in doing so demonstrated an algorithm that

takes only a composite number N as an input and determines with high probability a non-trivial

prime factor of N in a polynomial number of steps.

The n-qubit qft is easily defined. First, it is useful to convert from our prior notation of bit-

vectors x to the binary number representation x = x0 +2x1 + 4x2 + +2nxn. The qft is definedas

Uqft

2n1x=0

gx |x =2n1y=0

Gy |y , (1.8)

where G is the 2n-component vector of complex values calculated as the dft of the 2n-component

vector g.

Naively, it might seem to take 22n operations to compute the dft of the 2n complex amplitudes

gx , since for each of the 2n values Gy we must multiply all 2

n values of gx by phase factors and then

perform a summation. A step of critical importance in classical algorithms, however, was the 1965

discovery of the Fast Fourier Transform (fft) by Cooley and Tukey [12]. The fft exponentially

speeds up calculation of the dft to only order n2n operations without loss of accuracy. The critical

step for quantum algorithms was the discovery that the qft could be done in exponentially fewer

steps than even the fft, requiring only O(n2) operations! This does not mean that the qft can

replace the ubiquitous fft, as the qft is fundamentally quantum mechanical, operating in the full,

unmeasurable Hilbert space of n qubits.

10 Chapter 1. Introduction

The qft is useful for finding the eigenvalue of a unitary operator U corresponding to a known

eigenstate |u. I will now describe the algorithm for doing this starting from the answer. Since U isunitary, the desired eigenvalue has the form exp(2i), where 0 < 1. Let us represent witht-bit accuracy as x/2t. The algorithm seeks to generate the t-qubit state |x. The qft of |x is

Uqft |x = 12t

2t1y=0

e2ixy/2t |y = 1

2t

2t1y=0

e2iy |y .

This state may be generated by starting with one register of qubits in the superposition state

2t/2

y |y, obtained with Hadamard transformations as in the Deutsch-Josza algorithm, and asecond register of qubits in the known eigenstate |u. The desired state results from the operation

|y |u |yUy |u . (1.9)

The algorithm using this operation and the inverse qft to find is known as the phase estimation

algorithm. This is the quantum mechanical part of Shors algorithm.

The rest of Shors algorithm is classical number theory. To summarize, suppose we wish to find

the m factors of the number N . A classical computer can efficiently eliminate the simple possibilities

that N is even or N = ab for integers a and b; let us assume that N is neither. A random integer

x < N is chosen; a classical computer can quickly check to see that x and N are coprime. Then, the

unitary operator U used for the phase estimation algorithm is modular multiplication by x. That is,

for an input ket |y, U outputs the ket corresponding to x multiplied by y, modulo N . The operationcorresponding to Eq. (1.9) is therefore modular exponentiation, and the specific implementation of

it may be by a variety of classical (but reversible!) subalgorithms. Which subalgorithm takes the

least amount of time depends on the resources available and the size of N , as recently summarized

by Van Meter and Itoh [13]. It may readily be shown that the modular multiplication operator U

has r eigenstates |us with eigenvalues exp(2is/r), where the integer r satisfies xr = 1 mod N .Here the integer s runs from 0 to r 1. The number r is the order of x modulo N and it isthe desired output of the phase estimation algorithm. Although calculation of the eigenstates |uswould require knowledge of r, it may also be shown that the equal superposition of these states

yields the binary-representation ket |1. Using this known state instead of one of the |us statesin the phase estimation algorithm outputs a fraction s/r with a random s. If s/r is known with

sufficient precision, the denominator r may be efficiently calculated. With probability 1 2m, thisr is even and xr/2 6= 1 mod N , and if this is the case, then the greatest common denominator of

1.2 Theoretical Quantum Computing 11

LN

FS

Shor

C, 1 H

z

Shor

C, 1 H

z

Shor E

, 1 Hz

Shor E

, 1 Hz

Shor

C, 1 M

Hz

Shor

C, 1 M

Hz

Shor E

, 1 MH

z

Shor E

, 1 MH

z

1 second

1 minute

1 hour

1 month

1 year

100 years

100 1000 10,000

Figure 1.1: A comparison of execution times of classical and quantum factoring algo-rithms vs. the size L, in bits, of the number to be factored. The curve labelled nfsrepresents the expected scaling of the classical nfs algorithm with the total scale set bythe current world-record largest implementation of it, in which 104 personal computersrunning in parallel factored a 576-bit number in one month in 2003, according to rsaSecurity, Inc. The curves labelled Shor C represent an implementation of Shors quan-tum algorithm in which logical qubits may only couple to nearest neighbors. The curveslabelled Shor E represent an implementation in which any logical qubit may couple toany other logical qubit. These implementations are described by Van Meter and Itoh[13]. Both algorithms use 100L logical qubits. For comparison, clock speeds of 1 Hzor 1 MHz are shown; these clock speeds must include the time for extra processing forquantum error correction.

N and either xr/2 + 1 or xr/2 1 is a factor of N . If r is odd, or xr/2 = 1 mod N , the algorithmfails and should be repeated until it succeeds.

Most of this algorithm may be efficiently performed on a classical computer; the key quantum

mechanical steps are to feed a linear superposition state into the modular exponentiation operator

and the inverse qft. Details of both these quantum aspects and the classical aspects of the algorithm

may be found in Nielsen and Chuang [14].

Shors algorithm is the proving ground for quantum computation. Since the fastest known al-

gorithm for factoring on a classical computer, the number field sieve (nfs), is super-polynomial

in number of steps, and Shors algorithm is polynomial, a quantum computer seems to offer dra-

12 Chapter 1. Introduction

matic speed-up over classical computers, challenging the strong Church-Turing Thesis3. Figure 1.1

compares the speed of potential quantum computers for factoring n-bit numbers against optimized

networks of classical computers as a function of n. More practically, however, the slow speed

at which classical computers factor numbers is a critical assumption of the heavily used Riven-

Shamir-Adleman (rsa) encryption system [15]. This system is frequently used for security in digital

communications today, and a working quantum computer would compromise that security. Shors

algorithm has therefore extended the interest in quantum computers from physics and computer

science academia to defense and financial sectors.

1.2.3 Other Algorithms

Shors algorithm is currently the most important quantum algorithm, but it is by no means the only

one. The quantum Fourier transform has other applications; these generalize to finding hidden

subgroups of a group on which the function implemented by a unitary operation is defined. Solving

the discrete logarithm problem is another example of an algorithm in this class.

Another important algorithm is Grovers algorithm [16], which searches an unstructured list

for a target bit string with polynomial speed-up over classical search algorithms. Although the

improvement in computation time is not as dramatic as in the case of Shors algorithm, the ubiquity

of searching gives this algorithm its importance.

Specific algorithms have also been devised for using quantum computers to simulate other quan-

tum systems of interest, as proposed by Feynman. This is potentially one of the most important

uses for quantum computers in the future of technological development. A long-term dream is to

build a device that allows quantitative prediction for systems such as biological macromolecules,

high-temperature superconductors, and even atomic scale classical computers.

1.2.4 Quantum Error Correction and Fault Tolerance

None of these quantum algorithms will be useful unless the reality of decoherence is suppressed.

Quantum computers would not have any experimental promise if they were limited by the decoher-

ence processes observed for even the most long-lived laboratory qubits. Fortunately, Shor and others

showed soon after the development of Shors algorithm that errors in quantum systems (including

decoherence) can be corrected, even if the tools used to correct them are themselves faulty!

The continuous Hilbert space in which quantum states live would seem to require error correction

3It is important to note that a more efficient classical algorithm for factoring could exist; extensive efforts by greatminds to find it or prove its nonexistence have, however, met no success.

1.2 Theoretical Quantum Computing 13

similar to that needed for analog signals, which is notoriously more difficult than digital error

correction. However, just as the act of measurement stochastically digitizes the quantum state

of the quantum computer, so may it digitize a continuous set of errors, and then error correction

techniques similar to those for classical digital error processing are available.

The simplest example to demonstrate this point is a 3-qubit bit-flip code. Begin by considering

a classical digital computer, where it is possible for a bit to erroneously flip with small probability

< 1. A simple error correction technique is to copy this bit two times, and then periodically

correct any possible errors by examining the three copies and taking a majority vote. Then the

probability of error is reduced to the probability of two simultaneous bit flips, which, if the errors

are uncorrelated, has much smaller probability 2.

We would encounter two problems if we attempted to naively implement this on a quantum

computer. The first is the no-cloning theorem, which tells us that we cannot simply copy the

unknown state of a qubit to another qubit. The no-cloning theorem is very simple to prove, but its

introduction to quantum theory has been a crucial advance for understanding the nature of quantum

information. The proof is by contradiction. Suppose a quantum cloning device C existed, which

could take an arbitrary, unknown quantum state | and a second, initialized system |0 and copythe unknown state into that second system, i.e.

C | |0 = | | . (1.10)

Already we see that C cannot possibly be unitary; the information about the initial state of the

second system has been lost. As long as the cloning works, however, we will allow C to increase

total entropy. But we may also see that C cannot be linear, for if we apply it to | = | we obtain| |, and if we apply it to | = | 6= | we obtain | |. If C is linear then applying it to| = | + | yields | | + | |, which is different from the desired | |. Therefore such acloning device cannot exist in quantum mechanics, where allowable operations are non-negotiably

linear. This no-cloning theorem prevents us from simply copying a qubit in an arbitrary state and

taking a majority vote4.

The second problem we would encounter in carrying our classical majority-vote bit-flip code over

to quantum mechanics is the fact that the error might not be a full bit flip, but rather a partial bit

flip. A qubit is equivalent to a spin-1/2 particle with angular momentum ~I; this angular momentum

may point along any direction of the Bloch sphere. For example, the state of qubit j may begin as

4The no-cloning theorem does not prevent us from generating an ensemble of qubits in the same state; this may bedone by initializing every qubit in a known state and processing them identically. Ensemble-based quantum computingin this sense exhibits a simple form of majority-vote error correction.

14 Chapter 1. Introduction

Figure 1.2: A quantum circuit representing the 3-qubit bit-flip code. The qubit repre-sented by the top line is encoded into three qubits with two controlled-not operationson two ancilla qubits. A single qubit error occurs somewhere in the dashed box. Thesyndrome and recovery operations are implemented with two controlled-not operationsand two polarization measurements; the top qubit is flipped if both ancilla are measuredin state |1.

|1, with density operator = 1/2 + Izj , but the small error process, rather than flipping the qubitto the other pure eigenstate |0, simply depolarizes the qubit to some degree yielding the densityoperator = 1/2 + (1 2)Izj , for unknown < 1. We can describe this error process as

(1 )+ 4Ixj Ixj . (1.11)

It would seem at first that we need an error correcting procedure that restores the qubit for any

continuous value of ! This plays the same role as the bit-flip probability in the classical digital

case only if we projectively measure the polarization of our qubit, which we do not wish to do if we

want to maintain its coherence for quantum information processing.

Shor [17] pointed out that these two problems are fairly easily overcome. Rather than copying

our unknown qubit, we entangle it with two extra ancillary qubits. This may be done by applying

controlled-not operations to each ancillary qubit, which flip the ancillary qubit only if the unknown

qubit is in state |1, converting the state( |0 + |1

)|00 to |000 + |111. The quantum

circuit describing this operation is shown in Fig. 1.2. The error process then occurs on each qubit

individually, taking

= |2| |000000|+ |2| |111111|+ |000111|+ |111000|

1.2 Theoretical Quantum Computing 15

to

|2|((1 3) |000000|+ |100100|+ |010010|+ |001001|

)

+|2|((1 3) |111111|+ |011011|+ |101101|+ |110110|

)

+((1 3) |000111|+ |100011|+ |010101|+ |001110|

)

+

((1 3) |111000|+ |011100|+ |101010|+ |110001|

). (1.12)

The key thing to notice about this quantum state is that all new terms are orthogonal to the

original density operator. These orthogonal states may therefore be projected out by a suitable

syndrome detection and recovery operation. In this case, the syndrome detection is accomplished

by attempting to disentangle the two ancillary qubits from the original qubits, using the same

controlled-not operations. This results in the state

|2|((1 3) |000000|+ |111111|+ |010010|+ |001001|

)

+|2|((1 3) |100100|+ |011011|+ |110110|+ |101101|

)

+((1 3) |000100|+ |111011|+ |010110|+ |001101|

)

+

((1 3) |100000|+ |011111|+ |110010|+ |101001|

). (1.13)

Note that the states in the original density operator are preserved, and all error terms have resulted

in a bit flip of one or both of the ancillary qubits. Moreover, if we have measured both of the

ancillary qubits flipped from |00 to |11, then we know a bit-flip has occurred on our initial qubit,which we may correct with a simple bit-flip gate. Our initial qubit is therefore recovered in all cases.

The reason our recovery from a continuous error is accomplished with a digital bit flip is the use of

projective measurement to collapse the Hilbert space occupied by our qubit to subspaces where the

qubit either completed its erroneous flip or did not undergo any error at all.

This procedure has completely eliminated error for the error model chosen in Eq. (1.11). Such

an error model might arise due to high-temperature spin-relaxation processes, where it is taken as

a discretization of a master equation of the form

d

dt =

j

(4Ixj I

xj

). (1.14)

16 Chapter 1. Introduction

In some small time t, the evolution of three qubits each evolving by this equation is iterated

(t+ t) = (t) +

t+tt

j

[4Ixj (t

)Ixj (t)]dt

(1 3t)(t) + 4tj

Ixj (t)Ixj + 8

2t2jk

Ixj Ikx(t)I

xj I

xk + . . . .

This resembles the error model we have corrected with = t, but we have neglected correlated

errors, which occur to order 2 in this master equation approach. Our three-qubit code fails to

protect against these terms. They may be ignored given a sufficiently fast error correction period

t. In the end, our quantum error correction process has performed the same as our classical digital

error correction, eliminating bit-flip errors that are first order in with the use of two extra qubits

to redundantly store the initial information.

This example discussed only one type of error, the longitudinal depolarization of individual qubits

as described by Eq. (1.14). For other types of error, different encodings, syndrome measurements,

and recovery operations are possible, and these may be concatenated to protect against many types

of error simultaneously. Shors original nine-qubit code concatenated the above three-qubit code,

protecting against longitudinal depolarization, with a similar three-qubit code in an orthogonal basis

to simultaneously protect against transverse depolarization of individual qubits. An independent

approach to quantum error correction by A. Steane [18] was also developed at about the same time.

More efficient codes can be derived using techniques from classical digital error correction. One of

these so-called Calderbank-Shor-Steane (CSS) codes [19, 20] uses only 5 qubits to correct against

the same errors as the 9-qubit Shor code. The useful stabilizer formalism [21] has been a crucial aid

in the development of codes.

Another type of protection against decoherence operates on a different principle [22, 23]. This

is the use of decoherence-free subspaces. This type of protection may be succinctly summarized

as the exploitation of the symmetry of the error model to encode qubits in subspaces that are

invariant to the error operation. For example, suppose the qubits are spin-1/2 magnetic dipoles in

an environment with a global fluctuating magnetic field. The symmetry group for global rotations

of the qubits admits multiple irreducible representations, some of which correspond to angular

momentum singlet states. For two spins, there is a one-dimensional singlet subspace corresponding

to the Bell state | |. As a one-dimensional subspace, this state undergoes no evolutionunder global rotations, and therefore the noisy global field cannot change it. If the fluctuating field

is strictly in the z-direction, a further invariant subspace for the two-qubit system is the m = 0

state of the triplet subspace, the Bell state |+ |; in this high symmetry case these two states

1.2 Theoretical Quantum Computing 17

may provide a decoherence-free qubit. If the fluctuating magnetic field has components in all spatial

directions, only the singlet-states are invariant to it; fortunately, among a large number n of spin-1/2

dipoles there is a large degeneracy of singlet states. To be precise, there are

n!

(n/2 + 1)!(n/2 1)! 2n1

n

singlet states, which is most of the states! Asymptotically, then, one may encode O(n logn)noiseless qubits. Decoherence-free subspaces have already proven to be a valuable error-protection

tool in a variety of experiments, especially with trapped ions [24, 25] but their applicability is

ultimately limited due to the requirements of symmetry in the noise model.

A questionable assumption in these quantum error correction schemes is that the resources they

require can be reliably collected without making the initial error probability worse. Suppose that

we hope to correct one type of error. This could occur due to an ongoing physical noise process but

could also be due to erroneous state preparation, faulty logic, or imperfect quantum measurement.

To correct this error, we must prepare more states, perform more logic, and make more quantum

measurements. The time to perform these additional resources or the ongoing imperfection of our

system means that in fixing the original error, we could make the problem worse by causing more

errors. What is truly required for large-scale quantum computation is fault tolerance, a construction

in which state preparation, quantum logic, and even quantum measurement are performed so that

each add a sufficiently small amount of error per use that the error-correcting protocol may be

concatenated with itself indefinitely to achieve an arbitrarily accurate quantum computer. It was

again Shor [26] that introduced this notion of fault tolerance into quantum computation, and a

substantial number of researchers have developed constructions to show that fault-tolerant quantum

computation is possible, provided the errors in state preparation, logic, and measurement fall below

threshold values.

Suppose that our controlled-not gate is faulty, so that a bit-flip error occurs on the target

qubit with probability every time we use the gate. If we did no error correction at all, then

every controlled-not gate in our algorithm would accumulate an error and accurate computation

would not be possible, so we decide to employ the 3-qubit code discussed above. We have shown

that this encoding corrects bit-flip errors to order 2. During the act of encoding of this code, we

use one controlled-not gate on each ancilla, and the syndrome measurement requires one more

controlled-not gate on each ancilla, so the very act of encoding introduces an error of order 2,

which is corrected to order (2)2. As long as 1/2 this may be acceptable, but perhaps wedesire even more accuracy. We could improve our error-correction by encoding our encoded qubits

18 Chapter 1. Introduction

one more time. This means 2 controlled-not gates for each of our encoded qubits. For this to be

done fault-tolerantly, we must make sure that errors dont propagate; if a physical controlled-not

gate causes only a single bit error with probability , then an encoded controlled-not gate should

not cause more than one error in the encoded bit. In particular, a controlled-not gate between

two encoded qubits in the 3-bit code can happen qubit-by-qubit, preventing correlated errors from

accumulating during the gate. The encoded controlled-not therefore introduces an error no larger

than the residual error for a bit flip on the encoded qubits, (2)2. The total residual error after two

levels of encoding is therefore 2[(2)2]2. After k levels of encoding, the residual error is (2)2k

/2, a

number which quickly vanishes as k increases, as long as is smaller than the threshold of 1/2. If

all controlled-not gates are performed fault-tolerantly on the encoded qubits, the total error after

N gates is this small number multiplied by N . In this case, we need 3k qubits and just as many

operations per controlled-not gate. The encouraging conclusion is that if we can tolerate a total final

error , then the number of physical qubits and logic gates needed grows only polylogarithmically

in N and 1/.

This simple example hides the daunting resources actually required for fault-tolerant quantum

computing. In our example, we supposed that our only error was the random bit flip during a

faulty controlled-not; in reality there can be many types of error occurring during many types of

operations. We also only discussed the controlled-not gate for the 3-qubit bit-flip code, a gate which

is easy to make fault tolerant. In reality, many quantum gates do not have a natural fault-tolerant

architecture, in the sense that a naive construction causes low-level errors to propagate excessively

among encoded blocks of qubits. Creating fault-tolerant gates may require large additional resources

in ancillary qubits, projective measurements, and low thresholds depending on the gate, the error

model, and the encoding. More realistic threshold values for typical error models and fault-tolerant

architectures are of order 105 or worse, with individual fault-tolerant procedures requiring more

qubits and physical logic gates than have been implemented in any actual experiment to date. I

offer a brief discussion of the physical resources required in Chapter 2; a reader interested in the

further details of the theory of quantum error correction and fault-tolerant quantum computation

is advised to begin with Chapter 10 of Nielsen and Chuang [14].

Despite the physical difficulty of reaching the threshold for fault-tolerant quantum computation,

the very existence of a threshold gives hope that quantum computers could be realistically scaled to

arbitrarily large size, even in the presence of inevitable decoherence processes. Were it not for the

developments in quantum error correction and fault tolerance, there would be no hope for quantum

computation.

1.3 Experimental Quantum Computing 19

1.3 Experimental Quantum Computing

Either it is possible to build a quantum computer, or it isnt.

This trite little sentence is more profound than it seems, and as such has motivated the research

I present in the remainder of this dissertation. The key point is that if we show that quantum

computers are possible, which we can only do by actually building one, then mankind will have

an information processing tool unlike any it has possessed before. However, if by some discovery

or theory we are able to conclusively show that quantum computers are not possible, then the

argument against them will add a crucial chapter to the development of quantum mechanics and

the foundations of physics.

But grim reality must not be forgotten. We are not currently able to build quantum computers,

but the barriers that prevent us from doing so are far from fundamental. Those who build quantum

computers from trapped ions have trouble with random magnetic fields; those who build them from

superconducting Josephson junctions have impurities in oxide layers; those who build them from

photons and beamsplitters have spurious reflections from the surfaces of their photodetectors. Any

physical apparatus used to build a quantum computer must be extremely clean and efficient, and

ordinary technical problems are currently the only clear obstacle between reality and a world of

quantum computers.

If progress is to be made, we can only attempt to chip away at this mountain of technical

problems. There are many possible paths to the quantum computer, and until one succeeds we

cannot know whether quantum mechanics will provide us with a tool for more and more complex

technology in the distant future, or whether new physics will be found along the way to show us why

we are doomed to macroscopic classicality. Either way, however, the technical problems solved in

the quest will provide valuable knowledge with unknown future benefit to both our understanding

of the universe and our ability to manipulate it.

This dissertation focusses on one class of paths toward quantum computing; these are paths

making use of nuclei in the solid state, in particular without the help of electron wave function

engineering. In terms of the amount of money and the number of researchers and published papers,

the efforts described here represent a quiet minority in the race to quantum computing, but no one

can easily say which efforts will bear the most important fruit in the final harvest of technologies

for quantum computers.

In the following chapter I describe the needed resources for quantum computation, and then

without much technical detail the various ideas for meeting these resources in quantum computing

20 Chapter 1. Introduction

research around the globe. In Chapter 3 I focus on technologies for quantum computers using solid-

state nuclei. The advantage of nuclei are their extremely long relaxation times, and I discuss the

origins of nuclear relaxation processes in ensuing chapters, especially in silicon which has particular

promise for quantum computing architectures. In Chapter 4 and Chapter 5 I describe an experi-

mental exploration of the decoherence properties of silicon nuclei. Relaxation processes in nmr are

outlined in Chapter 6 and applied to silicon in Chapter 7. The notation used in all of these chapters

is explained in Appendix A.

The focus of the material in this thesis is on physical theory and experiment toward quantum

computing. There is little further discussion of algorithms, error correction, or communication

protocols. In other words, this work deals strictly with the hardware, and leaves the software

to others. Readers interested in algorithms and protocols are invited to begin with Nielsen and

Chuangs textbook on the subject [14].

Chapter 2

Physical Resources for Quantum

Computing

Somewhere around the place Ive got an unfinished short story about Schrodingers Dog;

it was mostly moaning about all the attention the cat was getting.

Terry Pratchett

2.1 Three Essential Requirements for Fault Tolerant Quan-

tum Computation

There is much room for creativity in the construction of quantum computing proposals. DiVincenzo

[27] has summarized the physical implementation of quantum computers and has listed five require-

ments for a quantum computer, although even these as originally stated may be too restrictive, and

are certainly not independent. In this chapter, I regroup these into three requirements, and focus

on the one which is especially challenging for nmr-based approaches to quantum computation.

2.1.1 An exponentially large and coherent Hilbert space, without expo-

nentially large space, time, or energy.

The easiest and by far most common example of a large Hilbert space is a collection of separate

qubits. A set of N qubits each with energy separation ~0 and effective volume V can be assembled

22 Chapter 2. Physical Resources for Quantum Computing

in volume NV and energy N~0. Even if these qubits are arranged in a one-dimensional line and

are only allowed to interact with their nearest neighber, the time required for universal quantum

computation (that is, the time to implement sufficiently many logic gates to approximate any desired

unitary operator on the qubits to a desired accuracy) is polynomial in N . However, the number

of physical logical gates required to allow fault-tolerant logic among arbitrary encoded qubits is

so large that fault-tolerant thresholds are extremely low for such one-dimensional, nearest neighbor

geometries; an architecture in which arbitrarily distant qubits can rapidly interact is much preferred.

Physical qubits are not absolutely essential. Quantum information processing can in principle

be performed with n-state systems instead of 2-state systems, or even with continuous quantum

variables. The only important consideration is that multiple quanta are a critical component of

establishing an exponentially large Hilbert space. As an example that abandons this principle,

suppose we had sufficient control to access the infinity of bound states of the hydrogen atom; for

an n-qubit computation we use 2n of its infinite states. The energy is clearly bound, since none of

these states exceed the 13.6 eV binding energy. This is certainly a large Hilbert space contained in

non-exponential energy. However, the effective volume of the atom grows exponentially, as the 2nth

state has roughly a radius of 2n Bohr radii. If we used the energy states of an electron trapped

in a hard box, the size would always be limited, but the energy would grow exponentially. These

requirements have been quantified by Blume-Kohout et al. [28].

Of course, decoherence can also l