FRAME VERSION WITH INDEX AND JPEGs (in case you do not see the frames)

Mladen Pavicic

University of Zagreb, Croatia

Consulting Editor: D. R. Vij

Library of Congress Cataloging-in-Publication Data

Pavicic, Mladen.

Quantum computation and quantum communication : theory and experiments

Mladen Pavicic.

p. cm.

Includes bibliographical references.

ISBN 10 0-387-24412-3

ISBN 13 978-0387-24412-9

ISBN 0-387-28900-2 (e-book)

1. Quantum computers 2. Quantum theory. I. Title

QA76.889.P38 2005 2005051725

© 2006 Springer Science+Business Media, Inc.

All rights reserved. This work may not not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

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Dedicated to the reader.

Dedication v

Preface ix

Acknowledgments xi

Introduction xiii

1. BITS AND QUBITS: THEORY AND ITS IMPLEMENTATION 1

1.1 The Turing Machine vs. a Computing Machine 1

1.2 Definition of a Turing Machine 2

1.3 Turing Computability 4

1.4 Bit Computability: Boolean Algebra 7

1.5 Bit Implementation: Transistors and Their Limits 9

1.6 Irreversible Bits: Logic Gates 12

1.7 Reversible Gates 14

1.8 Quantum Bits: Qubits 17

1.9 Flying Qubits and Circular Polarization 20

1.10 Superposition of Qubits 22

1.11 Bra-Ket Qubit Formalism 24

1.12 Operators 26

1.13 Detecting Qubits 27

1.14 Quantum Gates and Circuits 29

1.15 Qubit Computation and E-Business 31

1.16 Numbers and Bits 36

1.17 Entangled Qubits 39

1.18 General Single Qubit Formalism 45

1.19 Other Qubits and Universal Gates 51

1.20 Teleportation of Copies and the No-Cloning Theorem 56

1.21 Quantum Cryptography 64

1.22 Quantum Error Correction 72

1.23 Unconditional Security of Quantum Cryptography 81

2. EXPERIMENTS 87

2.1 Technological Candidates for Quantum Computers 87

2.2 Zeeman Effects 88

2.3 Liquid-State Nuclear Magnetic Resonance 94

2.4 Silicon-Based Nuclear Spins 99

2.5 Ion Traps 109

2.6 Future Experiments 123

2.7 Quantum Communication Implementation 125

3. PERSPECTIVES 135

3.1 Quantum Network 137

3.1.1 Laser 138

3.1.2 One-Atom Laser and Atom-Cavity Coupling 139

3.1.3 Single Photons on Demand 140

3.1.4 Laser Dark States 142

3.1.5 Cavity Dark States 144

3.1.6 Dark-State Teleportation 146

3.1.7 Quantum Repeaters 151

3.2 Quantum--Classical Coupling 159

3.2.1 Interaction-Free Computation 159

3.2.2 Kochen--Specker Setups 167

3.3 Quantum Algorithms 173

3.3.1 Quantum Coin---Deutsch's Algorithm 173

3.3.2 Deutsch-Jozsa and Bernstein-Vazirani Algorithms 176

3.3.3 Shor's Algorithm 180

3.3.4 Quantum Simulators 186

3.4 Quantum Turing Machines vs. Quantum Algebra 190

REFERENCES 199

INDEX 211

ON THE AUTHOR 223

What facilitated all these breakthroughs is the fact that at the present stage of development of quantum computation and communication, we deal with elementary quantum systems consisting of several two-level systems. The complexity of handling and controlling such simple systems in a laboratory has turned out to be tremendous, but the basic physical models we follow and calculate for the systems themselves are not equally intricate. We could say that the theory of the field leads the experiments in a particular way-with each new model we put forward and apply in the laboratory, we also build up and widen the theory itself. Therefore, we cannot just proceed with assembling quantum computers and quantum networks. We also have to use mathematical models to understand the physics of each step on the road to our goal.

As a consequence, both mathematics and physics are equally essential for any approach in the field and therefore for this book as well. The mathematics used in the book is a tool, but an indispensable tool because the physics of quantum computation and communication theory and their experiments cannot be grasped without good mathematical models. When we describe an experiment many times, we may get used to it, but this does not mean we are more at home with the principles and models behind it. This is why I have chosen to make this book an interplay between mathematics and physics. The idea of the book is to present those details that are used the most often both in theory and experiment and to dispense with many inessential ones. Also, the book is not conceived as a textbook, at least not as a primary one, but more as a guide to a better understanding of theory and experiments by coming back to the same concepts in different models and elaborations. Clear physical ideas make any formalism easy.

Mladen Pavicic

The first prediction, a beloved opening of speeches and papers, was made by the head of the electromagnetic relay calculator at Harvard, Howard Aiken, in 1956: "If it should turn out that the basic logics of a machine designed for the numerical solution of differential equations coincide with the logics of a machine intended to make bills for a department store, I would regard this as the most amazing coincidence that I have ever encountered" [Anonymous, 1997]

The amazing "coincidence" did happen and happens more and more every day, tempting us to consider it a part of the history of computers that took its own unexpected course ("Only six electronic digital computers would be required to satisfy the computing needs of the entire United States," Howard Aiken said in 1947): a program and a machine, software and hardware, were interwoven at the beginning and then became more and more separated. At least it seems so when we look at the development of computer designs since Charles Babbage's 1840s Analytical Engine. A program on punched cards or tapes and a machine for which the specific cards were made look inseparable, in contrast to today's programs which we move throughout the World Wide Web and compile and execute on virtually any computer.

Yet Alan Mathison Turing (and also Alonzo Church, Stephen Cole Kleene, and Emil Post independently at the same time) had already proved in 1936 that the only possible course the history could have taken was the one it in fact took. Turing used what we now also cite often and call a

The second prediction is known as

And this "exponential" element is what is essential for our development and what quantum computers are about. Apparently everything underlying the development of technology and society grows exponentially: research, information, production and organization complexity, and above all, the costs of keeping pace. So only an exponential increase of our computational and processing power and an exponential decrease of computer cost per processed bit could support such a development. Therefore, Moore's law was been kept as a guideline in the computer industry in past three decades and it has supported a global development during this period.

Gates in today's computers are switched on and off by about 1000 electrons. In 2010, the exponential Moore's Law would require that only about 10 electrons do the job. Miniaturization cannot go much further than that. It is true that many other possible roads could still keep up the pace for a few more years: insulating layers can be reduced in their thickness from the present 25 atoms to 4 or 5 atoms (wires connecting transistors in a chip already occupy more than 25% of its space); computing power can be increased by designing processors so as to contain execution units that process multiple instructions within one cycle; processors can rely on parallel compiling technology and use innovative software; and finally, chips can eventually get bigger by using reversible gates to avoid overheating. Still, by 2020 or 2025 computing technology will hit the quantum barrier, and if we want to support the growth of our technology and science beyond that point in time, we need to find a substitute for exponentially rising classical computational power by then. Actually, the exponential increase of the clock speed of processors (CPUs) already became linear in 2002 (see Fig. 3.1, p. 135), and an extensive patching activity onto classical hardware and software is currently under way in order to compensate for this lack of an exponential increase in speed (see p. 136).

Now that both Wirth's and Moore's laws are coming to an end, we should draw a moral from them. Wirth's law taught us that classical hardware development has prompted ever new software, and Moore's law taught us that this hardware development has followed an exponential trend of speed, memory, and lately of number of processors (multiple cores, multiple processors, clusters). Such an approach to computation will apparently change completely in the quantum realm. Quantum hardware is exponential in itself, and if we eventually succeed in making functional scalable quantum computers, we will dispense with the need for a steadily growing quantum hardware development - to make a quantum computer faster means to scale it up linearly or polynomially. We will also dispense with writing ever new software for faster and faster hardware. Once developed, quantum software (quantum algorithms) will simply scale up as we scale - and therefore speed up - quantum hardware.

The "exponential" is built into quantum hardware from its very first

The target of these courses, seminars, and textbooks is to teach and familiarize students and scientists with this new field - in which new research projects will keep opening for decades to come - and to help integrate the theory and experiments of quantum computation and communication into a would-be quantum network implementation. The goal of the book in front of the reader is the same; however, it allows her or him to digest the field "by reading." That means that there will be no homework and no exercises. Instead, most of the required details are elaborated within the main body of the book, and a polynomial complexity of reading is intended, optimally in one run.

So, a few words about the reader. She or he is expected to be familiar with higher mathematics and the basics of physics - in particular, quantum physics. The reader could be any former student who graduated in the technical or natural sciences, although an undergraduate student might also find many if not all sections of the book digestible. Students as well as specialists in the field might also find the nutshell approach of the book helpful and stimulating.

In 1936 several authors showed, in effect, that if a function is effectively calculable, then it is Turing computable and, of course, vice versa [Church, 1936c; Turing, 1936; Turing, 1937; Church, 1936a; Church, 1936b; Kleene, 1936; Post, 1936]. Turing concluded:

We do not need to have an infinity of different machines doing different jobs. A single one will suffice. The engineering problem of producing various machines for various jobs is replaced by the office work of "programming" the universal machine to do these jobsThis statement does not mean that Turing envisioned the "universal computer" we have today, although he was well acquainted with the project of breaking the cryptographic codes of German messages carried out on the Colossus (the British "computer" at Bletchley Park, which operated from 1943 until the 1950s). His

Turing machines, or any equivalent mathematical algorithms, are essential in order to decide whether a chosen problem is calculable or not, but we do not use them to write down a new program for, say, 3D modeling or speech recognition. Still, since there are many references to the Turing machine in the literature on quantum computing, let us provide some details [Hermes, 1969]. In doing so, we bear in mind that Turing machines and all related concepts are "concepts of pure mathematics. It is however very suggestive to choose a technico-physical terminology suggested by the mental image of a machine" [Hermes, 1969, p.31].

The Turing machine is neither today's "universal" computing machine - generally called a computer - nor a generator of new algorithms for the latter machine. Instead, it is simply a mathematical procedure to check whether a chosen algebra and/or calculus can or cannot be implemented into a computer. To show this, we present some details of the procedure. The details often appear in the literature without being put into the context of a final outcome and so are just left hanging, giving the impression of being building blocks for a computer, or an algorithm to be carried out on one. On the other hand, the notion of the classical Turing machine is rather important for understanding the role that the quantum Turing machine has in the theory of quantum computation.

A-gate, 103

adiabatic passage, 144

algebra

modular, 193

algorithm

Bernstein-Vazirani, 31, 179

Deutsch's, 31, 173

Deutsch-Jozsa, 31, 176

exponential speedup, 178

eigenvalue

exponential speedup, 186

exponential time, 186

Euclid's, 180

field sieve, 180

general number field sieve, 35

GNFS, 35, 180

Grover's, 31, 186

Kochen--Specker, 168

statistical exponential speedup, 171

MMP diagram, 168

Shor's, 31, 35, 36, 64, 98, 137, 180, 181

exponential speedup, 181

exponential time, 181

NMR, 98, 180, 183

Simon's, 31, 180

exponential speedup, 180

Alice

classical, 65

quantum, 68, 131

alkali-metal atoms, 147

all-optical, 124

CNOT gate, 153

ancilla, 78

AND

classical, 8

angular

momentum

electron, 91, 100

nuclear, 147

quantum number, 89, 147

total, 147

annihilation operator, 40, 59, 111

atom

interference, 163

lattice, 195

atom-cavity coupling, 140, 145

constant, 145

atomic

lattice, 195

atomic dipole matrix element, 114

B92 protocol, 81, 82

balanced function, 174

bandwidth, 33

barrier

oxide, 10

transistor, 10

BB84 protocol, 69, 82

BBN Technologies, 125

BCNOT gate, 153

beam splitter, 23

polarizing, 150

Bell

inequalities, 167

state, 43, 62, 154

decomposition, 156

Bennett-Brassard protocol, 69

Bernstein-Vazirani algorithm, 31, 179

binormality, 6

birefringent

plates, 129

prism, 40

bit, 1

train

Kane computer, 107

bit-flip, 72

correction, 80

quantum, 76

bits, 17

Bloch sphere, 51

blue sideband frequency, 121

Bob

classical, 65

quantum, 68, 131

Bohr magneton, 89

Boolean

algebra, xiii, 2, 7, 193

axioms for, 8

single axiom for, 9

circuit, 7

operation, 12

bra vector, 24

bra-ket notation, 20

bracket, 25

Calderbank-Shor-Steane code, 76

carrier frequency, 121

cat

Schrodinger, 57

cavity, 145

dark state, 146

optical, 138, 140, 163

spherical mirror, 140

QED, 88, 124

quantum electrodynamics, 88

CC-U gate

quantum, 55

CCNOT gate

classical

reversible, 16

quantum, 77

central processing unit, 14

cesium, 147

check matrix, 74

Church's thesis, 6

circuit

Boolean, 7

classical, 14

integrated, 135

reversible, 16, 54

CMOS, 10

NMOS, 10

PMOS, 10

quantum, 30, 32, 52, 55--57, 77--80, 122, 153, 176, 182, 191

diagram, 32, 182

interaction-free, 166

quantum logic, 61

size, 30

transistor, 12

circular polarization, 20

left-hand, 21

right-hand, 21

classical

circuit

integrated, 135

reversible, 16

cryptography, 65

logic, 194

completeness, 194

soundness, 194

clock speed

classical, xv, 135

cloned state, 58

closed subspace, 26

CNOT gate

all-optical, 153

bilateral, 153

classical

reversible, 15

f-, 174

interaction-free, 160

pseudo, 160

quantum, 55

ion trap, 122

Kane, 107, 108

NMR, 98

silicon-based spin, 107, 108

code, 73

codeword, 73

coherence

length

laser, 33

time

laser, 33

coincidence probability, 61, 62

collection efficiency, 128, 138

completeness

lattice, 195

complex numbers

field of, 191, 196

complexity

exponential, 35

subexponential, 35, 180

super-polynomial, 180

composite Hilbert space, 136

computer

human, 4, 6

computing

green, 15

physical, 34

tape, 3

continued fraction expansion, 185

continuous

quantum

computer, 194

variables, 194

continuous wave laser, 33

control qubit, 51

controlled-controlled-U gate

quantum, 55

controlled-controlled-NOT gate

classical

reversible, 16

quantum, 77

controlled-NOT gate

classical

reversible, 15

quantum, 55

cooling

laser

Doppler, 111

Sisyphus, 111, 119

copied state, 58

coprime, 180

coset, 84

Coulomb

potential, 195

repulsion, 110

countable orthonormal basis, 26

counterfactual computation, 166

coupler, 156

fiber, 81, 132

coupling constant, 98

CPU

classical, xv, 14, 135

quantum, 137

creation operator, 40

cryptography

classical, 65

quantum, 36, 64, 69

continuous variables, 126

entangled pairs, 126, 128

free space transmission, 126

phase-coding, 81

roadmap, 126

single-photon sources, 126, 127

weak laser pulses, 126

CSS code, 76

CW laser, 33

dark

counts, 127

state, 143, 145, 147

cavity, 146

mixing angle, 143

teleportation, 147

DARPA quantum network, 125

decidability, 6

decoherence, 123

delay

gate, 38

demand

photons on, 127

density

matrix, 47

operator, 28

detection, 27

deterministic

transition function

Turing machine, 4

Turing machine, 4

Deutsch's algorithm, 31, 173

Deutsch-Jozsa algorithm, 31, 176

exponential speedup, 178

diagram

MMP, 168

diode laser, 126

dipole

approximation, 114, 116

matrix element

atomic, 114

moment

electric, 145

magnetic, 89

Dirac's bra-ket notation, 20

discrete Fourier transform, 173

distributive lattice, 193

distributivity, 193

divergenceless field, 21

DiVincenzo Criteria, 124

donor

phosphorus, 102

dopant, 9

Doppler laser cooling, 111

down-conversion, 128

type-I, 130

type-II, 130

Earnshaw's theorem, 110

eavesdropping

quantum, 70

edge, 168

eigenfunction, 26

eigenket, 26

eigenvalue, 26, 186

algorithm

exponential speedup, 186

exponential time, 186

eigenvector, 26, 186

Einstein-Podolsky-Rosen pair, 44

electric dipole moment, 145

electric-field vector, 18

electron

angular

momentum, 100

angular momentum, 91

commutation relations, 117

magnetic

moment, 100

Planck energy, 93

single

transistor, 12, 105

spin, 91, 100

empty

state, 40

entangled

pair, 44

phase-coding, 132

polarization-coding, 131

photons, 42, 62

qubits, 57

states, 45, 57, 79

on demand, 122

entanglement, 20, 62, 63, 68, 167

EPR pair, 44

phase-coding, 132

polarization-coding, 131

equations

lattice, 197

error

correction

classical, 72

Hadamard gate, 77

quantum, 76

weigh, 75

Euclid's algorithm, 180

Euler angles, 48

evaluation according to rule, 5

Eve

quantum, 70, 132

exponential

complexity, 34, 35

improvement

quantum repeater, 159

speed increase, 135

speedup, 39

Deutsch-Jozsa algorithm, 178

eigenvalue algorithm, 186

Shor's algorithm, 181

Simon's algorithm, 180

statistical speedup

Kochen--Specker algorithm, 171

time, 34, 35

eigenvalue algorithm, 186

Shor's algorithm, 181

extraordinary ray, 68

f-CNOT gate, 174

Fabry-Perrot resonator, 138

factoring a number, 34, 180

fault-tolerant computation, 80

Fermi

operator, 117

fiber

coupler, 81, 132

fictitious magnetic

dipole

ion trap, 117

field

ion trap, 117

field

divergenceless, 21

irrotational, 21

longitudinal, 21

of complex numbers, 191, 196

of quaternions, 191, 196

of real numbers, 191, 196

sieve algorithm, 180

transversal, 21

finite dimensional space, 26

Fock

space, 40

state

single-photon, 127

Fourier

transform, 186

Fourier transform

discrete, 173

quantum, 173, 183

free space transmission

cryptography

quantum, 126

frustrated total internal reflection, 161

FTIR, 161

full adder, 37

gas

van der Waals, 190

gate

A, 103

BCNOT, 153

CNOT

all-optical, 153

bilateral, 153

classical reversible, 15

interaction-free, 160

ion trap, 122

Kane, 107, 108

NMR, 98

quantum, 55

silicon-based spin, 107, 108

delay, 38

Hadamard, 32, 175, 177, 186

ion trap, 121

NMR, 98

J, 103, 104

logic

classical, 12

quantum, 29

reversible, 15

NOT

classical, 11

ion trap, 121

quantum, 29

square root of NOT, 23, 29

NMR, 98

phase, 32

quantum, 17, 23, 29, 108

reversible

classical CNOT, 15

universal, 15

S, 107

Toffoli, 16, 54

universal

quantum, 53

reversible, 15

gcd, 180

general

number field sieve algorithm, 35, 180

recursiveness, 5

generator matrix, 49

GNFS algorithm, 35, 180

Godel

numbers, 6

numbering of Turing machines, 6

greatest common divisor, 180

green computing, 15

Grover's algorithm, 31, 186

gyromagnetic factor, 46, 90

Hadamard gate, 32, 175, 177, 186

error correction, 77

ion trap, 121

NMR, 98

half adder, 37

half-wave plate, 22, 149, 166

Hamiltonian

harmonic oscillator, 111

ion trap, 116

Jaynes-Cummings, 139

Kane computer, 104

local, 186

NMR, 97, 100

Hamming

code, 73, 76

codeword, 76

distance, 73

rule, 73

scheme, 73

harmonic oscillator, 111

Hamiltonian, 111

Heisenberg microscope, 164

Hermitian

conjugate operator, 26

operator, 27

hidden variable theory, 167

Hilbert

lattices, 195

space, 26

n-dimensional, 167

composite, 136

embedding, 156

polarization, 156

HWP, 22, 166

hyper-entangled quantum state, 156

hyperfine interaction, 100, 103, 147

hyperthreading technology, 136

ID Quantique, 125

idler photon, 129

information theory

quantum, 61

Intel, 135

interaction

picture, 120

strong, 190

weak, 190

interaction-free

CNOT gate, 160

experiment, 160

quantum

circuit, 166

resonance detection, 163

interferometer

Mach--Zehnder, 29, 34, 81, 132

Internet, 35, 65

interval analysis, 171

inverted population, 138, 140

ion trap, 88, 109

computer, 121

scalable, 122

fictitious magnetic dipole, 117

fictitious magnetic field, 117

Hamiltonian, 116

laser beam, 113

irrotational field, 21

isomorphism-free generation of MMP diagrams, 169

J-coupling, 98

J-gate, 103, 104

Jaynes--Cummings

Hamiltonian, 139

model, 116

join

lattice, 192

Jones vectors, 19, 21

Kane

CNOT gate, 107

computer, 103, 106

bit train, 107

Hamiltonian, 104

radio frequency magnetic field, 103

spin subspace, 106

magnetic

field, external, 101

Karnaugh map, 17

ket vector, 20, 24

key

distribution, 81

RSA, 35

sifted, 71

Kochen--Specker

algorithm, 168

statistical exponential speedup, 171

set, 168

setup, 167

theorem, 167

vectors, 168

KS

algorithms, 168

set, 168

setup, 167

vectors, 168

lambda-definability, 5

Lamb--Dicke

limit, 119

parameter, 119

Lande factor, 92

Larmor frequency, 93

resonance, 103

laser, 138

beam

atom interaction, 113

ion trap, 113

beam-electron-phonon interaction, 116

beam-ion interaction, 116

coherence length, 33

coherence time, 33

continuous wave, 33

cooling, 111, 119

diode, 126

phase-coding, 81

Doppler cooling, 111

one-atom, 139

pulse

quantum cryptography, 126

pump beam

down-conversion, 130

pumping, 63

lattice, 96, 192

atom, 195

completeness, 195

distributive, 193

equations, 197

Hilbert, 195

join, 192

meet, 192

modular, 193

operation, 192

orthocomplementation, 193

orthomodular, 193

superposition, 195

principle, 195

least significant bit, 37

left-hand circular polarization, 21

linear

ion trap, 110

optical elements, 31, 156

subspace, 27

linearization of wave equation, 46

linewidth, 33

local Hamiltonian, 186

logic

classical, 194

completeness, 194

soundness, 194

gate

classical, 12

quantum, 23, 29

reversible, 15

propositional, 7

quantum, 15, 61

completeness, 194

proper, 194

soundness, 194

reversible, 15

longitudinal field, 21

LSB, 37

mu-recursiveness, 5

Mach--Zehnder interferometer, 29, 34, 36, 81, 132

MagiQ Technologies, 125

magnetic

dipole moment, 89

fictitious dipole

ion trap, 117

fictitious field

ion trap, 117

field

Kane, external, 101

radio frequency, Kane, 103

radio frequency, NMR, 96

Zeeman, external, 88, 90

Zeeman, inner, 91

moment

electron, 100

nuclear, 100

quantum number, 89

magneto-optical trap, 163

Malus law, 19, 131

matrix

check, 74

density, 47

generation, 74

MatrixExp, 50, 98

measurement

quantum, 27

meet

lattice, 192

microchannel plate detector, 164

mixing angle

dark state, 143

MMP diagram, 168

algorithm, 168

isomorphism-free generation, 169

modular

algebra, 193

lattice, 193

modularity, 193

modulo, 8, 74, 78, 174, 181

momentum

electromagnetic, 21

space, 189

monolithic total-internal-reflection resonator, 161

Moore's Law, xiv, 135

MOSFET, 9

NPN, 9

PNP, 9

most significant bit, 37

MOTIRR, 161

MSB, 37

multicore technology, 136

NAND

classical, 8

negative absorption, 138

neutral atom, 124

NMOS, 9

NMR, 88, 124, 180

square root of NOT gate, 98

computer, 98, 99

radio frequency magnetic field, 96

Hadamard gate, 98

Hamiltonian, 97, 100

Shor's algorithm, 98, 180, 183

no-cloning theorem, 58

nonlinear optics, 128

NOR

classical, 8

normal algorithm, 6

NOT

classical, 8

gate

classical, 11

ion trap, 121

quantum, 29

square root of NOT gate, 23, 29

NMR, 98

NPN MOSFET, 9

nuclear

fusion, 87

magnetic

moment, 100

magnetic resonance, 88, 124

spin quantum number, 94

number

field sieve

general algorithm, 35

operator, 111

state

notation, 117

one-atom laser, 139

operation

Boolean, 12

lattice, 192

operator, 26

adjoint, 26

annihilation, 40, 59

creation, 40

density, 28

Fermi, 117

Hermitian conjugate, 26

linear, 26

projection, 27

unitary, 27, 186

optical

cavity, 138--140, 163

spherical mirror, 140

element

linear, 31, 156

path difference, 34

resonator, 138

OR

classical, 8

order, 181

ordinary ray, 68

ortho-isomorphism, 196

orthoautomorphism

unitary, 196

orthocomplementation, 193

ortholattice, 192

orthomodular

lattice, 193

poset

sigma, 193

orthomodularity, 193

orthonormal basis

countable, 26

oxide

barrier, 10

P-doped substrate, 9

parametric

down-conversion, 128

generation, 128

parity, 72

bit, 72

path difference

optical, 34

Paul trap, 109

Pauli

matrices, 46

problem, 156

uniqueness, 156

Penning trap, 109

phase

gate, 32

retarder, 22

shift, 18, 76

correction, 80

phase-coding

entangled pair, 132

EPR pair, 132

quantum cryptography, 81

phonon, 110, 112

anticommutation relations, 112

state, 112

phosphorus donor, 102

photon

angular momentum, 20

entangled, 62

gun, 127

idler, 129

on demand, 127, 140

particle aspect, 18

Planck energy, 18

pump, 129

signal, 129

total angular momentum, 21

wave aspect, 18

physical computation, 34

pickup coil, 96

Planck energy, 18

electron, 93

photon, 18

PMOS, 9

PNP MOSFET, 9

Poisson distribution, 127

polarization, 18

circular, 20, 147

Hilbert space, 156

linear, 18, 19, 22, 39, 40, 44, 59, 63, 68, 70, 129, 147

nonlinear, 129

polarization-coding

entangled pair, 131

EPR pair, 131

polarizing beam splitter, 150

population

inversion, 138

inverted, 140

poset

orthomodular

sigma, 193

Poynting vector, 19, 21

precession, 96

frequency, 97

prime

relatively, 67, 180

privacy amplification, 85

probabilistic

device

quantum, 20

intrinsically

quantum measurement, 27

transition function

Turing machine, 190

Turing machine, 190

probability

amplitudes

quantum, 27

coincidence, 61, 62

quantum, 20, 27, 28

projector, 27

propagation vector, 20

propositional logic, 7

protocol

B92, 81, 82

BB84, 69, 82

Bennett-Brassard, 69

six-state, 81

pseudo CNOT gate, 160

public

key cryptography

classical, 66

key protocol

classical, 67

RSA, 67

pump photon, 129

pure state, 28, 40

purification, 153

QED, 90

cavity, 88, 124

QKD, 125, 127

QND device, 157

quantum

bit-flip, 76

bits, 17

CCNOT gate, 77

circuit, 30, 32, 52, 55--57, 77--80, 122, 153, 176, 182, 191

diagram, 32, 182

interaction-free, 166

quantum logic, 61

size, 30

coin, 173

computation

roadmap, 124

computer

cavity quantum electrodynamics (QED), 88

continuous, 194

ion trap, 121

Kane, 103

NMR, 98

silicon-based, 103

controlled-controlled-NOT gate, 77

cryptography, 36, 64, 69

continuous variables, 126

entangled pairs, 126, 128

free space transmission, 126

phase-coding, 81

roadmap, 126

single-photon sources, 126, 127

weak laser pulses, 126

dot, 128, 152

Fourier transform, 173, 183

gate, 17, 23, 29, 108

information theory, 61

key distribution, 125

logic, 15, 61

completeness, 194

gate, 17, 29

proper, 194

soundness, 194

measurement, 27

intrinsically probabilistic, 27

network, 20

DARPA, 125

nondemolition detection device, 157

number

angular momentum, 89, 147

magnetic, 147

probabilistic device, 20

probability, 20, 27, 28

amplitudes, 27

register, 182

repeater, 151

transition function

Turing machine, 190

Turing machine, 190

quarter-wave plate, 22, 149, 166

quaternions

(skew) field of, 191, 196

qubit, 19

control, 51

entangled, 57

flying, 20, 123

stationary, 20, 123

target, 51

qutrits, 51

QWP, 22, 166

Rabi

flopping frequency, 115

frequency, 115, 119

radio-frequency (RF), 96, 103

magnetic field

Kane computer, 103

NMR computer, 96

Raman

adiabatic passage

stimulated, 144

scheme, 113

ray

extraordinary, 68

ordinary, 68

real numbers

field of, 191, 196

reckonability, 5

red sideband frequency, 121

reflectance, 41

reflection coefficient, 41

reflectivity, 161

register

quantum, 182

relatively prime numbers, 67, 180

resonance

interaction-free detection, 163

resonant pulses, 96

resonator, 161

Fabry-Perrot, 138

optical, 138

total-internal-reflection

monolithic, 161

reversible

circuit, 16, 54

logic, 15

logic gate, 15

universal gate, 15

RF, 96, 103

coil, 96

electric field, 110

field, 103

generator, 96

pulses, 96

signals, 96

right-hand circular polarization, 21

Ritt's characteristic set calculations, 171

roadmap

quantum computation, 124

quantum cryptography, 126

rotating wave approximation, 115

round-trip, 161

RSA

key, 35

public key protocol, 67

rubidium, 148, 165

S-gate, 107

sigma-orthomodular poset, 193

scalable

ion trap computer, 122

Schrodinger

cat states, 57

equation, 114, 136, 142, 187, 197

picture, 120

selection frequency, 162

semiconductor, 9

Si, 103

separable subspace, 26

SET, 12, 105

Sheffer stroke, 7

Shor's algorithm, 31, 35, 36, 64, 98, 137, 180, 181

exponential speedup, 181

exponential time, 181

NMR, 98, 180, 183

Si semiconductor, 103

Si substrate, 103

sideband frequency, 121

sifted key, 71

signal photon, 129

silicon-based nuclear spins, 88, 100

Simon's algorithm, 31, 180

exponential speedup, 180

single-electron transistor, 12, 105

single-photon

detector, 126

Fock state, 127

singlet state, 43, 62

maximal, 153

nonmaximal, 153

Sisyphus cooling, 111, 119

six-state protocol, 81

size of a quantum circuit, 30

SO(3) group, 47

solid state, 124

space

finite dimensional, 26

Fock, 40

Hilbert

n-dimensional, 167

composite, 136

embedding, 156

polarization, 156

spin, 48

vector, 24

special

2-dimensional unitary group, 47

orthogonal 3-dimensional rotation group, 47

spherical mirror

optical cavity, 140

spin

electron, 91, 100

space, 48

subspace

Kane computer, 106

spin-orbit interaction, 91, 102, 147

state

Bell, 154

decomposition, 156

cloned, 58

copied, 58

dark, 143, 145, 147

cavity, 146

teleportation, 147

empty, 40

entangled, 45, 57, 79

on demand, 122

Fock

single-photon, 127

hyper-entangled, 156

metastable, 138

number, 117

pure, 28, 40

singlet, 43, 62

maximal, 153

nonmaximal, 153

teleported, 64

triplet, 62, 131

unknown, 58, 137

vacuum, 40, 117

stimulated Raman adiabatic passage, 144

STIRAP, 144, 146, 149

Stokes

laser, 144

beam, 143

field, 144

strong

interaction, 190

SU(2) group, 47

subexponential

complexity, 35, 180

subspace

closed, 26

linear, 27

separable, 26

spin

Kane computer, 106

substrate, 9

Si, 103

super-polynomial complexity, 180

superconducting, 124

superposition, 20, 22, 24, 167

lattice, 195

principle

lattice, 195

swapper, 156

syndrome, 75

tally, 4

target qubit, 51

teleportation, 61, 63, 68, 150, 151

deterministic, 122

teleported state, 64

tensor product, 40

Toffoli gate, 16, 54

tokamak, 87

total-internal-reflection resonator

monolithic, 161

transistor, 9

barrier, 10

channel, 9

circuit, 12

single electron, 12, 105

transition function

deterministic

Turing machine, 4

probabilistic

Turing machine, 190

quantum

Turing machine, 190

transmission coefficient, 41

transmittance, 41

transversal field, 21

trapped ion, 109, 124

computer, 109

trial divisions, 34

trichloroethylene, 97

triplet state, 62, 131

truth table, 8

tunneling

resonator, 161

Turing machine, 1

deterministic, 4

transition function, 4

Godel numbering of, 6

probabilistic, 190

transition function, 190

quantum, 190

transition function, 190

universal, 7

Turing-computable, 5

type-I down-conversion, 130

type-II down-conversion, 130

unconditional security of quantum cryptography, ix, 72, 83, 126

unitary

orthoautomorphism, 196

unitary operator, 27, 186

universal

quantum gate, 53

reversible gate, 15

Turing machine, 7

unknown state, 58, 137

vacuum

state, 40, 117

van der Waals gas, 190

vector

space, 24

vertex, 168

wave

approximation

rotating, 115

equation

linearization, 46

vector, 20

weak

interaction, 190

Weierstrass solid immersion lenses, 128

welcher Weg, 164

which way, 164

winner (would be quantum computer), 124

XOR, 8, 65, 72, 74, 85, 174

Zeeman

effect

anomalous, 88

normal, 88

nuclear, 95

magnetic

field, external, 88, 90

field, inner, 91

Hamiltonian part, 104

splitting, 90, 142

sublevels, 147

ISBN 0-387-24412-3

p. 22, the second line above Eq. (1.17) should read:

p. 114, the line below Eq. (2.67) should read:

p. 114, 2nd line from bottom should read:

p. 115, the line above Eq. (2.68) should read:

p. 136, 11th line from the top should read:

p. 139, the first half of the 8th line from bottom should read:

p. 147, Eq. (3.19) should read:

p. 147, Eq. (3.20) should read:

p. 147, the 5th line from bottom should read:

p. 148, the 1st line of the caption of Figure 3.6 should read:

p. 148, the 9th line from bottom should read:

p. 149, Eq. (3.25) should read:

p. 151, Eq. (3.33) should read:

p. 166, the 3rd line of the caption of Figure 3.16 should read:

p. 166, the 5th line of the caption of Figure 3.16 should read:

p. 167, the last line of the 2nd paragraph should read:

p. 197, Eq. (3.127) should read: