This chapter discusses the complex numbers and to develop fluency in the manipulation. A knowledge of complex numbers is essential for understanding quantum computation. The set of all complex numbers, C, together with the operations of addition and multiplication as defined above form a mathematical structure called a field. The derivation requires the use of Maclaurin series which we discuss first. Maclaurin series expansions are a way of representing functions as an infinite sum of terms involving increasing powers of a variable x, say. Essentially, the statement that is a field means that we are able to add, subtract and multiply any complex number and we can divide by any non-zero complex number and still remain in the field.
This chapter discusses the complex numbers and to develop fluency in the manipulation. A knowledge of complex numbers is essential for understanding quantum computation. The set of all complex numbers, C, together with the operations of addition and multiplication as defined above form a mathematical structure called a field. The derivation requires the use of Maclaurin series which we discuss first. Maclaurin series expansions are a way of representing functions as an infinite sum of terms involving increasing powers of a variable x, say. Essentially, the statement that is a field means that we are able to add, subtract and multiply any complex number and we can divide by any non-zero complex number and still remain in the field.
This chapter provides the vocabulary in a less than formal way, with more formal treatments being detailed. Quantum computation is multidisciplinary drawing as it does from computer science, physics and mathematics. Technicalities of a field need not concern us further, and whenever we refer to a field generally, it may be helpful for you to have the fields R or C in mind. Loosely speaking this means that the real numbers obey rules, known as field axioms, that enable us to add, subtract, multiply and divide by any non-zero number, in the manner in which we are already very familiar. The chapter describes practice using some preliminary techniques, such as modular arithmetic, working with Boolean connectives, and equivalence relations that will be important as you move through the book.
You, have you really understood all that stuff? What? The story of imaginary numbers? Robert Musil, The Confusions of Young Törless (1907) 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). Reader Tip . Many readers will find that they are already familiar with some of the material presented in this chapter. The reader who feels confident in her comprehension of the fundamental knowledge of complex numbers, the basic operations, and their properties can safely move on to later chapters. We suggest, though, that you at least skim through the following pages to see what topics are covered. Return to Chapter 1 as a reference when needed (using the index to find specific topics). 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.
Philosophy is written in that great book which continually lies open before us (I mean the Universe). But one cannot understand this book until one has learned to understand the language and to know the letters in which it is written. It is written in the language of mathematics, and the letters are triangles, circles and other geometric figures. Without these means it is impossible for mankind to understand a single word; without these means there is only vain stumbling in a dark labyrinth. Galileo Galilei 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. Reader Tip . The reader might find some of this chapter to be “just boring math.” If you are eager to leap into the quantum world, we suggest reading the first two or three sections before moving on to Chapter 3. Return to Chapter 2 as a reference when needed (using the index and the table of contents of find specific topics).
Quantum mechanics is based on linear algebra. The general theory uses infinite dimensional vector spaces. Fortunately for us, to describe spin or polarization we need only finite dimensions, which makes things much easier. In fact, we need only a few tools. At the end of this chapter I have given a list. The rest of the chapter explains how to use these tools and what the calculations mean. There are many examples. It is important to work carefully through all of them. The mathematics introduced here is essential to everything that follows. Like much mathematics, it can seem complicated when it is first introduced, but it becomes almost second nature with practice. The actual computations only involve addition and multiplication of numbers, along with an occasional square root and trigonometric function.
The mathematical formalism of quantum theory in terms of vectors and operators in infinite-dimensional complex vector spaces is very abstract. The definitions of many mathematical quantities used do not seem to have an intuitive meaning, which makes it difficult to appreciate the mathematical formalism and understand quantum mechanics. This book provides intuition and motivation to the mathematics of quantum theory, introducing the mathematics in its simplest and familiar form, for instance, with three-dimensional vectors and operators, which can be readily understood. Feeling confident about and comfortable with the mathematics used helps readers appreciate and understand the concepts and formalism of quantum mechanics. This book is divided into four parts. Part I is a brief review of the general properties of classical and quantum systems. A general discussion of probability theory is also included which aims to help in understanding the probability theories relevant to quantum mechanics. Part II is a detailed study of the mathematics for quantum mechanics. Part III presents quantum mechanics in a series of postulates. Six groups of postulates are presented to describe orthodox quantum systems. Each statement of a postulate is supplemented with a detailed discussion. To make them easier to understand, the postulates for discrete observables are presented before those for continuous observables. Part IV presents several illustrative applications, which include harmonic and isotropic oscillators, charged particle in external magnetic fields and the Aharonov–Bohm effect. For easy reference, definitions, theorems, examples, comments, properties and results are labelled with section numbers. Various symbols and notations are adopted to distinguish different quantities explicitly and to avoid misrepresentation. Self-contained both mathematically and physically, the book is accessible to a wide readership, including astrophysicists, mathematicians and philosophers of science who are interested in the foundations of quantum mechanics.
This chapter explores the rapidly developing field of quantum computing, which essentially devises the power to change science and technology. After an introductory outline, the fundamental ideas of quantum computing are covered, setting it apart from classical computing. The chapter then goes on to describe the features of a quantum computing system, highlighting its unique traits and possible advantages. The reliance on complex numbers and advanced mathematics is a central point of quantum computing [ 1 ]. Fundamental mathematical concepts are important to understand the quantum algorithms, which are explained in more detailed way. In quantum computing, the main idea is that quantum bits can have several states. The quality of quantum processing is extensively explored in this chapter. The chapter also covers a range of quantum computing techniques, highlighting their resourceful methods for tackling problems. Finally, the uses of quantum computing and its influence in many areas, such as optimization, materials research, and cryptography, are described and analyzed. This ample introduction will provide readers with thoughtful principles and the mathematical and practical significance of quantum computing.