Introduction to Quantum Computation and Information

The following sections are included:

• FOREWORD

• PREFACE

• CONTENTS

The marriage of quantum physics and information technology has the potential to generate radically new information processing devices. Examples are quantum cryptosystems, which provide guaranteed secure communication, and quantum computers, which manipulate data quantum mechanically and could thus solve some problems currently intractable to conventional (classical) computation. This introductory chapter serves two purposes. Firstly, I discuss some of the basic aspects of quantum physics which underpin quantum information technology (QIT). These will be used (and in some cases further expanded upon) in subsequent chapters. Secondly, and as a lead into the whole book, I outline some of the ideas of QIT and its possible uses.

For six decades, physicists have broken their heads over quantum entanglement. But by now we have learned to do more than break our heads over it. This review explains what is so baffling about entanglement and also what we can do with it. Entanglement is a resource for teleporting quantum states and constructing unbreakable codes, a resource that we can extract, purify, distribute and consume. The applications of entanglement lead us to develop new conceptual tools and adapt old ones—in particular, the concept of entropy, which helps us exploit entanglement efficiently. Here we define entanglement and show, via the Einstein-Podolosky-Rosen (EPR) paradox and Bell's inequality, how it implies quantum nonlocality. We draw a parallel between reversible heat engines and reversible transformations among entangled states. This parallel leads to the "entropy of entanglement" as the measure of entanglement of bipartite pure states. We also discuss the entanglement of density matrices.

We provide a general introduction to the concept of quantum information, discussing its relation to quantum measurement theory and properties of entanglement. We give an account of some of its basic features, notably quantum dense coding, quantum teleportation and the compression of quantum information. We include an explanation of the formalism of density matrices and the concept of von Neumann entropy, which play a fundamental role in much of quantum information theory.

The contest between code-makers and code-breakers has been going on for thousands of years. Recently, quantum mechanics has made a remarkable entry in the field. On the one hand, it has been rigorously proven that quantum cryptography can provide absolute security for communications between two users. On the other hand, code-breakers in possession of a quantum computer can easily break popular encryption schemes such as RSA and Data Encryption Standard (DES) which are essentially intractable with any classical computer. Here I survey these recent developments.

In this chapter different experimental Quantum Cryptography setups based on optical fibers are presented. Performance and technical problems are discussed and compared. The transmission distances and error rates are essentially limited by the photon counters that are presently available. Possible eavesdropping strategies, the use of single photon sources and open air experiments are briefly discussed.

When applied to computation, quantum mechanics opens completely new perspectives; by exploiting quantum mechanical features such as entanglement or super-position, one can solve some problems much more efficiently than with classical computers. One of the most striking examples is the factoring problem. This task is believed to be out of reach of classical computers as soon as the number of digits in the number to factor exceeds a certain limit. In this contribution, we present the basic notion of quantum computation and review the most common quantum algorithms discovered so far.

Quantum error correction (QEC) is a central component of quantum information theory. It is a powerful general method to restore the state of a quantum system after it has been subject to noise. The theory of quantum error correction is introduced, starting from the principles of error correction for classical communication channels. The treatment concentrates on the construction of quantum error correcting codes, and on the syndrome extraction operation, by which the quantum state is recovered after errors have occurred.

The discovery of quantum error correction has greatly improved the long-term prospects for quantum computing technology. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment, or due to imperfect implementations of quantum logical operations. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. In principle, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. It may be possible to incorporate intrinsic fault tolerance into the design of quantum computing hardware, perhaps by invoking topological Aharonov-Bohm interactions to process quantum information.

Practical quantum computing is discussed, from the viewpoint of using ion traps or applications of quantum optics. A linear ion trap quantum computer coupled to external laser beams is described in detail, as is an alternative realization based on cavity quantum electrodynamics. An experimentally feasible approach to quantum error correction for ion trap systems is given. The possibility of quantum computer networking, using optical fibres to couple systems together coherently, is also discussed.

Nuclear spins have long coherence times and are easily manipulated using radio-frequency pulses. Single molecules, with spin-spin couplings provided by electronic bonds, are thus attractive physical systems for performing quantum computation. However, observation of small nuclear induction signals typically requires a large number, , of molecules. This presents fundamental problems for quantum computation, which traditionally requires pure state inputs, and strong measurements. Quantum computation with nuclear magnetic resonance was made possible by the invention of techniques which create effective pure states from thermal mixtures, and algorithmic steps which make strong measurements unnecessary. The theoretical basis for quantum computation with bulk materials, and experimental results demonstrating implementation of simple quantum gates and circuits are described.

Classical information and computation theory are now naturally viewed as a subset of a more comprehensive theory concerned with the transmission and processing of intact quantum states. The new theory is broadly concerned with the possible transformations of quantum states into one another, often in a multiparty setting, where part of a possibly entangled quantum system is held by each of several observers. Many open questions remain concerning the kind and quantity of resources—including classical messages, shared entanglement, noiseless or noisy quantum communications channels, and (perhaps most fundamentally) direct physical interaction between the parties—that can be used to perform these transformations, and the extent to which various combinations of resources can be substituted for one another.