"This book arms to introduce the basics of quantum computing to advanced undergraduates anti beginning graduate students in electrical engineering arid computer science who have practically no background in quantum mechanics, but a fair level of mathematical sophistication. The basic ideas of quantum mechanics are developed in the context of two- and other finite state systems and illustrated by means of well-chosen physical examples. The heart of the book is devoted to a treatment of quantum gates and circuits, and following it, a discussion of quantum algorithms, including Shor's factoring algorithm and Grover's search algorithm. The purpose of the book is to make recent exciting developments in this emerging field accessible to the next generation of computer scientists as well as others without a traditional background in physics, End-of-chapter exercises, historical and biographical footnotes, a detailed glossary and a large number of references also make this a valuable text or reference work." — P.K. Aravind, Physics Department, *Worcester Polytechnic Institute*

"Quantum computing is a new promising area of research that investigates how the laws of quantum mechanics allow new forms of computation exponentially more efficient than any classical counterpart. This book aims at giving a gentle introduction to the basic concepts and mathematical techniques of this interdisciplinary research area to a readership with no previous background in quantum mechanics." — G. Massimo Palma, *University of Milano and INFM*

"Since it draws on same deed aspects of physics and mathematics, quantum computing can be a challenging subject to present. The Marinescus have done an admirable job of writing a highly readable and self-contained textbook on this important and fascinating new area of computer science. This is a thorough and very readable introduction to quantum computing. It should be especially useful as the text far an upper-level undergraduate or beginning graduate course aimed at computer science and engineering majors. The Marinescus have managed the difficult task of writing a book an quantum computing that is both technically complete and a pleasure to read. " — John Hayes, *University of Michigan*

RATIONALE

Tremendous progress has been made in the areas of quantum computing and quantum information theory during the past decade. Thousands of research papers, a few solid reference books, and many popular science books have been published in recent years on this subject. The growing interest in quantum computing and quantum information theory is motivated by the incredible impact these disciplines are expected to have on how we store, process, and transmit data and knowledge in this information age.

Computer and communication systems using quantum effects have remarkable properties. Quantum computers enable efficient simulation of the most complex physical systems we can envision. Quantum algorithms allow efficient factoring of large integers with applications to cryptography. Quantum search algorithms considerably speed up the process of identifying patterns in apparently random data. We can guarantee the security of our quantum communication systems because eavesdropping on a quantum communication channel can be detected with high probability.

It is true that we are years, possibly decades, away from building a quantum computer requiring little, if any power at all, filling up the space of a grain of sand, and computing at speeds that are unattainable today even by covering tens of acres of floor space with clusters made from tens of thousands of the fastest processors built with current state-of-the-art solid-state technology. All we have at the time of this writing is a 7-qubit quantum computer capable of computing the prime factors of a small integer, 15 139. To break a code with a key size of 1024 bits requires more than 3000 qubits and 108 quantum gates 82. Building a quantum computer presents tremendous technological and theoretical challenges. At the same time, we are witnessing a faster rate of progress in quantum information theory than in quantum computing. Applications of quantum cryptography seem ready for commercialization. Recently, a successful quantum key distribution experiment over a distance of some 100 km has been announced.

It is difficult to predict how much time will elapse from the moment of a great discovery until it materializes into a device that profoundly changes our lives or drastically affects our understanding of natural phenomena. The first atomic bomb was detonated in 1945, less than 10 years after the discovery of nuclear fission by Lise Meitner and Otto Hahn 91. The first microprocessor was built in late 1970s, some 30 years after the creation of the transistor on December 23, 1947 by William Shockley, John Bardeen, and Walter Brattain. Francis Harry Compton Crick and James Dewey Watson discovered the double helix structure of the genetic material in 1957 and the full impact of their discovery will continue to reverberate for years to come.

The authors believe that the time to spread the knowledge about quantum computing and quantum information outside the circle of quantum computing researchers and students majoring in physics is ripe. Students and professionals interested in information sciences need to adopt a different way of thinking than the one used to construct today's algorithms. This certainly presents tremendous challenges, since, for many years, computer science students have been led to believe that they can get by with some knowledge of discrete mathematics and little understanding of physics at all. We are going back to the age when a strong relationship between physics and computer science existed.

TOPICS, PREREQUISITES, AND CHAPTER DESCRIPTIONS

This text is devoted to quantum computing. We treat the quantum computer as a mathematical abstraction. Yet, we discuss in some depth the fundamental properties of a quantum system necessary to understand the subtleties of counterintuitive quantum phenomena such as entanglement. Chapter 1 introduces the reader to the quantum world by way of several experiments. Chapter 2 provides the most basic concepts of quantum mechanics and of the supporting mathematical apparatus. Chapter 3 introduces the qubit and hints at simple physical realizations of a qubit. Chapter 4 is devoted to quantum gates and quantum circuits. Chapter 5 presents quantum algorithms. The last chapter, Chapter 6, introduces the reader to quantum teleportation, quantum key distribution, and dense coding, and then presents reversible computations. The text is intended to be self-contained; concepts, definitions, and theorems from linear algebra necessary to develop the mathematical apparatus of quantum mechanics are introduced in Chapter 2. Appendix A introduces basic algebraic structures. Appendix B presents modular arithmetic necessary for understanding the factoring algorithms. Appendix C is devoted to the Walsh-Hadamard transform, and Appendix D summarizes basic concepts related to the Fourier transform. *Approaching Quantum Computing* is intended as a textbook for a one-semester first course in quantum computing. The time table we suggest for covering the material is: five weeks for Chapters 1 and 2, five weeks for Chapters 3 and 4, four weeks for Chapter 5 and Appendices A, B, C and D, and two weeks for Chapter 6. Any graduate or undergraduate student with a solid background in linear algebra, calculus, and physics should be able to do well in the class.

Features

This volume combines a qualitative presentation with a more rigorous, quantitative analysis. Whenever possible, we attempt to avoid the sometimes difficult mathematical apparatus, the trademark of quantum mechanics. In his marvelous book *A Brief History of Time* 71, Stephen Hawking, the astrophysicist, who is now the Lucasian professor, shares with his readers the warning he got from his editor: "Expect the sales to be cut in half for every equation in your book." There are *k*° – 102 equations in this series of lectures and 2100 ~ 100010 is a very large number. Detailed presentation and step-by-step analysis to illustrate the behavior of quantum circuits are given, along with numerous examples that will guide the reader in solving the problems at the end of each chapter. A solutions manual for instructors who adopt the book is available through the publisher.