NMR Quantum Information Processing (eBook)

Eduardo Azevedo, Tito Bonagamba, Jair C. C. Freitas, Roberto Sarthour Jr., Ivan Oliveira (Autoren)

eBook Download: EPUB

2011 | 1. Auflage
264 Seiten
Elsevier Science (Verlag)
978-0-08-049752-5 (ISBN)

Infos

Quantum Computation and Quantum Information (QIP) deals with the identification and use of quantum resources for information processing. This includes three main branches of investigation: quantum algorithm design, quantum simulation and
quantum communication, including quantum cryptography. Along the past few years, QIP has become one of the most active area of
research in both, theoretical and experimental physics, attracting students and researchers fascinated, not only by the potential
practical applications of quantum computers, but also by the possibility of studying fundamental physics at the deepest level of quantum phenomena.
NMR Quantum Computation and Quantum Information Processing describes the fundamentals of NMR QIP, and the main developments which can lead to a large-scale quantum processor. The text starts with a general chapter on
the interesting topic of the physics of computation. The very first ideas which sparkled the development of QIP came from basic considerations of the physical processes underlying computational actions. In Chapter 2 it is made an introduction to NMR, including the hardware and other experimental aspects of the technique. In
Chapter 3 we revise the fundamentals of Quantum Computation and Quantum Information. The chapter is very much based on the extraordinary book of Michael A. Nielsen and Isaac L. Chuang, with
an upgrade containing some of the latest developments, such as QIP in phase space, and telecloning. Chapter 4 describes how NMR
generates quantum logic gates from radiofrequency pulses, upon which quantum protocols are built. It also describes the important technique of Quantum State Tomography for both, quadrupole and spin
1/2 nuclei. Chapter 5 describes some of the main experiments of quantum algorithm implementation by NMR, quantum simulation and QIP in phase space. The important issue of entanglement in NMR QIP
experiments is discussed in Chapter 6. This has been a particularly exciting topic in the literature. The chapter contains a discussion
on the theoretical aspects of NMR entanglement, as well as some of the main experiments where this phenomenon is reported. Finally, Chapter 7 is an attempt to address the future of NMR QIP, based in
very recent developments in nanofabrication and single-spin detection experiments. Each chapter is followed by a number of problems and solutions.

* Presents a large number of problems with solutions, ideal for students
* Brings together topics in different areas: NMR, nanotechnology, quantum computation
* Extensive references

Inhalt

Front cover 1
NMR Quantum Information Processing 4
Copyright page 5
Preface 8
Acknowledgments 10
Contents 12
Brief Historical Survey and Perspectives 16
References 21
Chapter 1. Physics, Information and Computation 24
1.1 Turing Machines, logic gates and computers 24
1.2 Knowledge, statistics and thermodynamics 30
1.3 Reversible versus irreversible computation 33
1.4 Landauer's principle and the Maxwell demon 35
1.5 Natural phenomena as computing processes. The physical limits of computation 36
1.6 Moore's law. Quantum computation 39
Problems with solutions 42
References 46
Chapter 2. Basic Concepts on Nuclear Magnetic Resonance 48
2.1 General principles 48
2.2 Interaction with static magnetic fields 50
2.3 Interaction with a radiofrequency field - the resonance phenomenon 53
2.4 Relaxation phenomena 56
2.5 Density matrix formalism: populations, coherences, and NMR observables 59
2.6 NMR of non-interacting spins 1/2 62
2.7 Nuclear spin interactions 67
2.8 NMR of two coupled spins 1/2 76
2.9 NMR of quadrupolar nuclei 83
2.10 Density matrix approach to nuclear spin relaxation 88
2.11 Solid-state NMR 90
2.12 The experimental setup 94
2.13 Applications of NMR in science and technology 98
Problems with solutions 98
References 105
Chapter 3. Fundamentals of Quantum Computation and Quantum Information 108
3.1 Historical development 108
3.2 The postulates of quantum mechanics 110
3.3 Quantum bits 111
3.4 Quantum logic gates 112
3.5 Graphical representation of gates and quantum circuits 115
3.6 Quantum state tomography 119
3.7 Entanglement 121
3.8 Quantum algorithms 126
3.9 Quantum simulations 139
3.10 Quantum information in phase space 140
3.11 Determining eigenvalues and eigenvectors 145
Problems with solutions 146
References 150
Chapter 4. Introduction to NMR Quantum Computing 152
4.1 The NMR qubits 152
4.2 Quantum logic gates generated by radiofrequency pulses 155
4.3 Production of pseudo-pure states 168
4.4 Reconstruction of density matrices in NMR QIP: Quantum State Tomography 177
4.5 Evolution of Bloch vectors and other quantities obtained from tomographed density matrices 183
Problems with solutions 186
References 195
Chapter 5. Implementation of Quantum Algorithms by NMR 198
5.1 Numerical simulation of NMR spectra and density matrix calculation along an algorithm implementation 198
5.2 NMR implementation of Deutsch and Deutsch-Jozsa algorithms 200
5.3 Grover search tested by NMR 202
5.4 Quantum Fourier Transform NMR implementation 204
5.5 Shor factorization algorithm tested in a 7-qubit molecule 205
5.6 Algorithm implementation in quadrupole systems 208
5.7 Quantum simulations 209
5.8 Measuring the discrete Wigner function 214
Problems with solutions 216
References 219
Chapter 6. Entanglement in Liquid-State NMR 222
6.1 The problem of liquid-state NMR entanglement 222
6.2 The Peres criterium and bounds for NMR entanglement 224
6.3 Some NMR experiments reporting pseudo-entanglement 226
Problems with solutions 232
References 235
Chapter 7. Perspectives for NMR Quantum Computation and Quantum Information 236
7.1 Silicon-based proposals: solution for the scaling problem 237
7.2 NMR quantum information processing based on Magnetic Resonance Force Microscopy (MRFM) 241
7.3 Single spin detection techniques: solution for the sensitivity problem 246
7.4 NMR on a chip: towards the NMR quantum chip integration 249
Problems with solutions 251
References 256
Index 258

Textauszug

Brief Historical Survey and Perspectives

Various names are commonly associated to the invention and development of modern computing science. Among them, are George Boole (1815–1864), author of a work published in 1854 with the title: An investigation into the laws of thought, on which are founded the mathematical theories of logic and probabilities, which founded the nowadays called Boolean Algebra, and Claude Shannon (1916–2001) who, in 1938 on his MIT MSc Thesis, A symbolic analysis of relay and switching circuits, proposed a way for representing Boolean logic operators through relays and switches.

However, the Theory of Computation became an area of abstract mathematics only after the work of Alan Turing (1912–1954) and Alonzo Church (1903–1995). On his attempt to answer one of the challenges proposed by the great mathematician David Hilbert in 1928, the entscheidungsproblem or decision problem, Turing arrived to an abstract model of computation known as the Turing Machine. His idea was published in 1936 as a ground breaker paper entitled On computable numbers, with an application to the entscheidungsproblem [1]. A Turing Machine operates with a minimum number of symbols and instructions to perform logic operations: it is the embryo of all modern programmable computers.

Another breakthrough paper appeared twelve years afterwards, in 1948, again by Claude Shannon: A mathematical theory of communication [2]. On this paper, Shannon defined the unit of information, the binary digit, or bit,1 and established the theory which tells us the amount of information (i.e., the number of bits) which can be sent per unit time through a communication channel, and how this information can be fully recovered, even in the presence of noise in the channel. This work founded the Theory of Information.

The computation and information technologies have developed very close to each other, in an astonishingly rapidly pace, for the last 50 years. Nowadays, a few square centimeters computer chip possesses hundreds of millions of electronic constituents, and a hairy thin optical fibre can transmit and maintain millions of conversations simultaneously!

On the side of pure Physics, the 20th Century also produced some “miracles”, one of them – and possibly the most important of all – was Quantum Mechanics. The early development of this theory has attached to it a whole team of brilliant scientists: Max Planck, Niels Bohr, Albert Einstein, Louis de Broglie, Erwin Schrödinger, Wolfgang Pauli, Werner Heisenberg, only to name some of the best known. Quantum mechanics contains the rules of how to approach and solve problems involving particles such as electrons, protons, nuclei, atoms, molecules, and the interactions between these particles and radiation. Along the years, computers entered physics as a powerful ally for the analysis and development of physical models in particle and nuclear physics, condensed matter, gravitation, astrophysics, biological and ecological systems, and so on. In particular, the development of condensed matter magnetism and semiconductor physics resulted in important feedback to computer technology itself. This symbiotic relationship between physics and computers, deepened for decades until the point where computers themselves started to be seen by the physicists, no longer as an auxiliary tool for the solution of complicated mathematical problems, but as physical systems, subject to the laws of physics, just like everything else! This insight led to a novel and exciting area of research in Physics: Quantum Computation and Quantum Information.

Quantum Information is the area of research in physics in which quantum resources are identified for the application in information processing, as well as the means to produce, store, send and recover information traveling through communication channels. One example of quantum resource for communication is entanglement, and one example of quantum information processing is superdense coding. To the more specific application of quantum resources to the development of quantum computer algorithms and quantum hardware, we call Quantum Computation. One example of quantum algorithm is the Shor factorization algorithm, and one example of quantum computing hardware are nuclear spins.

The “formal” beginning of the research field called Quantum Computation and Quantum Information can be attributed to a paper published in 1980 by Paul Benioff [3]: The computer as a physical system: a microscopical quantum mechanical Hamiltonian model of computers as represented by Turing machines. In this paper it is pointed out for the first time that unitary transformations undergone by quantum systems can be used to implement computing logical operations. However, the work of Benioff was inspired by an earlier paper, published in 1973 by the IBM physicist Charles Bennett [4]. In his paper, Logical reversibility of computation, Bennett showed that computation could be built entirely on the basis of reversible logic, although actual computers operate with irreversible processes. Indeed, computation is carried out in computers through the action of the so-called logic gates. One complete set of such gates are the NOT, AND and OR gates. Whereas NOT is a reversible gate (in the sense that the information at the input of the gate can be recovered applying the gate to the output), AND and OR are irreversible, in the sense that information is lost in their action, implying an increase of entropy equal to at least kB ln 2 for each bit which is lost.2 On the other hand, quantum unitary transformations are reversible: from the knowledge of the state of a quantum system in time t0, one can obtain the state in later time t: |ψ(t) = U(t, t0)|ψ(t0), where U(t,t0) is a unitary propagator which satisfies the Schrödinger equation. However, since UU† = 1, where 1 is the identity matrix, one can recover |ψ(t0) from |ψ(t) through the operation: |ψ(t0) = U† (t, t0)|ψ(t). Of course, this is only valid for isolated systems. One of the major triumphs of Quantum Information Theory has been the development of tools which allow the treatment of non-isolated systems for quantum computation.

After Benioff, in the year of 1985, David Deutsch gave a decisively important step towards quantum computers presenting the first example of a quantum algorithm [6]. The Deutsch algorithm shows how quantum superposition can be used to speed up computational processes. Another influent name is Richard Feynman, who was involved about the same time in the discussions of the viability of quantum computers and their use for quantum systems simulations [7].

However, it was in 1994 that a main breakthrough happened, calling the attention of the scientific community for the potential practical importance of quantum computation and its possible consequences for modern society. Peter Shor discovered a quantum algorithm capable of factorizing large numbers in polynomial time [8]. Classical factorization is a kind of problem considered by computation scientists to be of exponential complexity. This basically means that the amount of time required to factorize a number N bits long, increases exponentially with N. In contrast, a quantum computer running Shor algorithm would require an amount of time which would be a polynomial function of N. This is a huge difference! To give an example, if N = 1024 bits, a classical algorithm would take about 100 thousand years to factorize the number, whereas Shor algorithm would accomplish the task in a few minutes!

Shor algorithm has not yet been tested in numbers that long, but its quantum working principles have already been demonstrated in laboratory, through the technique of nuclear magnetic resonance (NMR) [9]. The algorithm clearly raises important concerns about the security of cryptosystems based on the factorization of large numbers, such as the RSA protocol. Arthur Eckert captures the essence of the problem in the quote [10] “… modern security systems are in a sense already insecure… ”.

A few years after the discovery of Shor algorithm, in 1997, another important algorithm was discovered by Lov Grover [11]. The so-called Grover algorithm is a quantum search algorithm, which makes use of quantum superposition and quantum phase interference to find an item in a disordered list of N items with a squared speedup with respect to an equivalent classical algorithm. After the discoveries of Shor and Grover algorithms the interest in quantum computation and quantum information has grown dramatically along the years, as exemplified in Figure 1, which shows the number of refereed papers published in the subject from 1990 till nowadays.3

Figure 1 Number of papers published on quantum information and quantum computation...

Produkt-Details

• Erscheint lt. Verlag: 18.4.2011
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Naturwissenschaften ► Chemie ► Analytische Chemie
• Themenwelt: Naturwissenschaften ► Chemie ► Physikalische Chemie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Angewandte Physik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• ISBN-10: 0-08-049752-7 / 0080497527
• ISBN-13: 978-0-08-049752-5 / 9780080497525