This bibliographic essay reviews seminal papers in quantum computing. Although quantum computing is a young science, its researchers have already published thousands of noteworthy articles, far too many to list here. Therefore, this appendix is not a comprehensive chronicle of the emergence and evolution of the field but rather a guided tour of some of the papers that spurred, formalized, and furthered its study.

Quantum computing draws on advanced ideas from computer science, physics, and mathematics, and most major papers were written by researchers conversant in all three fields. Nevertheless, all the articles described in this appendix can be appreciated by computer scientists.

READING SCIENTIFIC ARTICLES

Do not be deterred if an article seems impenetrable. Keep in mind that professors and professionals also struggle to understand these articles, and take comfort in this epigram usually attributed to the great physicist Richard Feynman: “If you think you understand quantum mechanics, you don't understand quantum mechanics.”

Some articles are difficult to understand not only because quantum theory is devilishly elusive but also because scientific writing can be opaque. Fortunately, there are techniques for tackling scientific articles, beginning with these preliminary steps:

1. ▪ Read the title. It may contain clues about the article's purpose or findings.

2. ▪ Read the abstract. It summarizes the article and will help you recognize important points when you read them.

3. ▪ Read the introduction and conclusion. Usually in plain language, the introduction and conclusion will help you decode the rest of the article.

4. […]

In Chapters 1 and 2 we developed the necessary mathematical apparatus and terminology that will be used throughout this book. Chapter 3 has provided some heuristics and gently led us to the threshold of quantum mechanics. It is now time to open the door, introduce the basic concepts and tools of the trade, and continue our journey to quantum computing.

In Section 4.1 we spend a few words on the motivations behind quantum mechanics. We then introduce quantum states and how they are distinguishable from one another through observations. Section 4.2 describes observable physical quantities within the quantum framework. How observable quantities are measured is the topic of Section 4.3. The dynamics of quantum systems, i.e., their evolution in time, is the focus of Section 4.4. Finally, in Section 4.5, we revisit the tensor product and show how it describes the way in which larger quantum systems are assembled from smaller ones. In the process, we meet the crucial notion of entanglement, a feature of the quantum world that pops up again in the chapters ahead.

QUANTUM STATES

Why quantum mechanics? To answer this question, we have to hearken back in time to the dawn of the twentieth century. Classical mechanics still dominated the scene, with its double-pronged approach: particles and waves.

Quantum computing is a fascinating new field at the intersection of computer science, mathematics, and physics, which strives to harness some of the uncanny aspects of quantum mechanics to broaden our computational horizons. This book presents some of the most exciting and interesting topics in quantum computing. Along the way, there will be some amazing facts about the universe in which we live and about the very notions of information and computation.

The text you hold in your hands has a distinct flavor from most of the other currently available books on quantum computing. First and foremost, we do not assume that our reader has much of a mathematics or physics background. This book should be readable by anyone who is in or beyond their second year in a computer science program. We have written this book specifically with computer scientists in mind, and tailored it accordingly: we assume a bare minimum of mathematical sophistication, a first course in discrete structures, and a healthy level of curiosity. Because this text was written specifically for computer people, in addition to the many exercises throughout the text, we added many programming drills. These are a hands-on, fun way of learning the material presented and getting a real feel for the subject.

In this chapter we are going to describe quantum programming, i.e., the art and science of programming a quantum computer. In Section 7.1, we briefly sketch what it means to program a quantum computing device. Section 7.2 covers a simple version of quantum assembler, based on the so-called QRAM architecture. Section 7.3 describes possible steps leading to higher-level programming languages and constructs. We conclude this chapter with Section 7.4, a short discussion on quantum emulators.

PROGRAMMING IN A QUANTUM WORLD

As you are about to read this chapter, you have undoubtedly been exposed to computer programming in a variety of flavors, and are perhaps already an accomplished programmer of real-life applications. Programming a classical machine carries an immediate, unambiguous sense. However, we are going to leave the familiar world of binary chips, and learn how to program some as yet unspecified quantum hardware.

In a sense, theoretical computer science is uniquely qualified to study quantum computing. After all, Alan Turing and the other founders of theoretical computer science studied formal computation long before engineers actually produced a real-life computer. At present, large-scale quantum computers are not a reality yet. Nevertheless, the theoretical analysis of quantum computability and complexity is well on its way.

In Section 8.1, we start with a quick review of some of the basics of deterministic and nondeterministic Turing machines and the complexity classes that they engender. However, we shall discuss them in a way that is easily generalizable for our purposes. Section 8.2 moves on to probabilistic Turing machines and their zoo of complexity classes. Our main objective is found in Section 8.3, where we meet quantum Turing machines and their complexity classes. We shall also state some basic theorems and ideas about quantum computation.

DETERMINISTIC AND NONDETERMINISTIC COMPUTATIONS

Theoretical computer science deals with the question, “What is computable?” We must immediately qualify the question: “computable according to which model of computation?” It turns out that if we omit the question of efficiency, all sufficiently complicated formal models of computation can simulate each other. However, in order to fix our ideas and notation, we have to stick with one and work with it.

The topic of this chapter is quantum information, i.e., information in the quantum world. The material that is presented here lies at the core of a variety of areas within the compass of quantum computation, such as quantum cryptography, described in Chapter 9, and quantum data compression. As quantum information extends and modifies concepts developed in classical information theory, a brief review of the mathematical notion of information is in order.

In Section 10.1, we recall the basics of classical information theory. Section 10.2 generalizes this work to get quantum information theory. Section 10.3 discusses classical and quantum data compression. We conclude with a small section on errorcorrecting codes.

CLASSICAL INFORMATION AND SHANNON ENTROPY

What is information really? In the mid-forties, an American mathematician named Claude Shannon set out to establish a mathematical theory of information on a firm basis. To see this, we use Alice and Bob. Alice and Bob are exchanging messages. Let us say that Alice can only send one of four different messages coded by the letters A, B, C, and D.

This book covers many major developments in quantum computing, but the field is still young, and there will no doubt be many more developments in the future. These future developments will include research discoveries, of course, but they will also include trends in industry, surges in media coverage, and tides of public interest. This appendix describes tools that can help you track quantum developments of all kinds.

KEEPING ABREAST OF POPULAR NEWS

There are scores of newspapers, magazines, and other popular news sources, any one of which might run a story about the newest quantum development. How will you know if one does? You can keep an eye on your favorite news sources, but you will miss many stories that way. A better tactic is to use a news aggregator, such as Google News (http://news.google.com/), which allows you to search current and past stories from a multitude of news sources. You can add Google News to your stable of frequently visited sites, but the most efficient way to use it is to set up an alert or RSS feed and let the news come to you. After you perform a Google News search that yields good results, simply click “Alerts” to set up an alert that will notify you by e-mail of new stories that satisfy your search. Alternatively, click “RSS” to set up an RSS feed that will deliver those stories directly to your RSS reader.

In this chapter we explore the merging of quantum computation and classical cryptography. This is a new and exciting field of pure and applied research known as quantum cryptography.

We begin with the basics of classical cryptography in Section 9.1. Section 9.2 demonstrates a quantum cryptographic protocol that uses two different bases. We improve on this in Section 9.3, where a protocol with one basis is employed. Section 9.4 shows how to use entanglement to secretly send a message. We conclude with Section 9.5, in which teleportation is demonstrated.

CLASSICAL CRYPTOGRAPHY

Before delving into quantum cryptography, we need to familiarize ourselves with the core ideas of classical cryptography. A good place to start is the following definition.

Definition 9.1.1 Cryptographyis the art of concealing messages.

Indeed, this is precisely what the etymology reveals: “Cryptography” is a compound of two Greek words, crypton and graphein, which mean, respectively, hidden and writing.

Turning an ordinary message into an indecipherable one is called encryption. The opposite action, i.e., restoring the original message, is decryption. The original message is generally referred to as the plaintext, and the encrypted message is the ciphertext. A method for encryption is often referred to as an encryption protocol.

Although its history is relatively recent, quantum computing is already, by all standards, a very broad area of research. There is simply no way we can cover anything more than a relatively small fraction of its interesting topics. There are also many fascinating themes that are not exactly part of quantum computing per se, but are nevertheless closely related to it.

In this appendix, we list a number of suggestions for further exploration, covering some items we omitted from our text. It is our sincere hope that students will find them useful for their presentations or perhaps inspire them to select others that they can discover on their own.

Note to the student: Now it is your turn! The best way to really learn something is to teach it. There is no substitute for spending hours preparing a lecture and getting ideas straight so that you can present them. Knowing that other people will be asking you questions and learning from you will force you to understand the material at a deeper level.

You are urged to choose a subject from an area that you find interesting. Much time and energy is going to be spent learning, understanding, and preparing, so you might as well enjoy your choice from the start.

Complex numbers lie at the very core of quantum mechanics and are therefore absolutely essential to a basic understanding of quantum computation. In this chapter we present this important system of numbers from both the algebraic and the geometric standpoints. Section 1.1 presents some motivation and the basic definitions. The algebraic structure and operations on complex numbers are given in Section 1.2. The chapter concludes with Section 1.3, where complex numbers are presented from a geometric point of view and advanced topics are discussed. Our hope is that this chapter will help you get a little closer to what Sir Roger Penrose has very aptly called the “magic of complex numbers” (Penrose, 2005).

BASIC DEFINITIONS

The original motivation for the introduction of complex numbers was the theory of algebraic equations, the part of algebra that seeks solutions of polynomial equations.

Now that we have the mathematical and physical preliminaries under our belt, we can move on to the nuts and bolts of quantum computing. At the heart of a classical computer is the notion of a bit and at the heart of quantum computer is a generalization of the concept of a bit called a qubit, which shall be discussed in Section 5.1. In Section 5.2, classical (logical) gates, which manipulate bits, are presented from a new and different perspective. From this angle, it is easy to formulate the notion of quantum gates, which manipulate qubits. As mentioned in Chapters 3 and 4, the evolution of a quantum system is reversible, i.e., manipulations that can be done must also be able to be undone. This “undoing” translates into reversible gates, which are discussed in Section 5.3. We move on to quantum gates in Section 5.4.

BITS AND QUBITS

What is a bit?

Definition 5.1.1Abitis a unit of information describing a two-dimensional classical system.

There are many examples of bits:

1. ▪ A bit is electricity traveling through a circuit or not (or high and low).

2. ▪ A bit is a way of denoting “true” or “false.”

3. […]

Quantum theory is cast in the language of complex vector spaces. These are mathematical structures that are based on complex numbers. We learned all that we need about such numbers in Chapter 1. Armed with this knowledge, we can now tackle complex vector spaces themselves.

Section 2.1 goes through the main example of a (finite-dimensional) complex vector space at tutorial pace. Section 2.2 provides formal definitions, basic properties, and more examples. Each of Section 2.3 through Section 2.7 discusses an advanced topic.

In this chapter a compendium of all the schemes discussed is given, along with the source of the scheme in the argumentation literature. We have tried to make the compendium most useful to the reader by presenting the schemes that represent the most commonly used forms of argument, including not only those used in everyday discourse, but also certain schemes that are important in legal and scientific reasoning. Many of these schemes have subtypes, and research on the classification of the subtypes, and even more generally, research on determining which schemes are subspecies of other schemes, is not yet at an advanced stage. Thus while we have made occasional remarks on these matters, hoping to provide some insight, we have not generally listed all the known subspecies of the schemes. There are two especially important exceptions that need to be noted.

We have included a fairly comprehensive account of the known subschemes of the argument from popular opinion, to give the reader an idea of how it can be important to recognize many of these different subschemes in dealing with common arguments of the type associated with informal fallacies. The other important exception is the case of the argumentum ad hominem. Many subschemes for this type of argument have been recognized, and much work on trying to organize and classify them has been conducted (Walton, 1998).