Abstract With this chapter, we begin to transit from digital information processing to quantum information processing. However, to be able to master this transition, we first need to familiarize ourselves with quantum mechanical principles and the mathematics used to describe them. Since we already know about abstract state spaces, tensor products, and Hamiltonian operators and their eigenvectors and eigenvalues, this should not be too difficult or too alien. And yet, there remain a few challenges: Mathematical descriptions of quantum states usually involve the Dirac notation, require complex linear algebra in high dimensional spaces, and encode unintuitive probabilistic interpretations. Rising to these challenges, we briefly study the notion of Hilbert spaces, introduce the Dirac notation, and discuss the axioms of quantum mechanics and some of their consequences. All this will admittedly be dense and abstract but prepare us for everything that is still to follow. Our main goal is to emphasize that quantum mechanics and therefore quantum computing is just complex linear algebra in very high dimensional spaces so that quantum computing needs a different kind of thinking than we are used to in digital computing.
Abstract With this chapter, we begin to transit from digital information processing to quantum information processing. However, to be able to master this transition, we first need to familiarize ourselves with quantum mechanical principles and the mathematics used to describe them. Since we already know about abstract state spaces, tensor products, and Hamiltonian operators and their eigenvectors and eigenvalues, this should not be too difficult or too alien. And yet, there remain a few challenges: Mathematical descriptions of quantum states usually involve the Dirac notation, require complex linear algebra in high dimensional spaces, and encode unintuitive probabilistic interpretations. Rising to these challenges, we briefly study the notion of Hilbert spaces, introduce the Dirac notation, and discuss the axioms of quantum mechanics and some of their consequences. All this will admittedly be dense and abstract but prepare us for everything that is still to follow. Our main goal is to emphasize that quantum mechanics and therefore quantum computing is just complex linear algebra in very high dimensional spaces so that quantum computing needs a different kind of thinking than we are used to in digital computing.
Abstract Quantum mechanics is mathematically formulated in terms of a Hilbert space, which is a vector space with a defined inner product. Linear algebra is the study of vector spaces and the (linear) operations thereon, and so it follows that having a solid grasp of the basics of linear algebra is essential for studying quantum mechanics and, therefore, quantum computation. All the linear algebra prerequisites necessary to understand the theoretical results presented in the remainder of the book are detailed in this chapter. The chapter also acts as a nomenclature reference for the book, by providing all the notations that are used thereafter.
Quantum mechanics is like a pot: it is almost indestructible and extremely rigid, but also very flexible because you can use any ingredients for your soup. Göran Lindblad Quantum mechanics Although our first four chapters have been very mathematical, quantum mechanics has never been very far away. Let us recall how the mathematical structure of quantum mechanics is usually summarized at the end of a first course in the subject. First of all the pure states are given by vectors in a Hilbert space. If that Hilbert space is finite dimensional it is simply the vector space equipped with a scalar product of the particular kind that we called a Hermitian form in Eq. (3.82). Actually a pure state corresponds to an entire equivalence class of vectors; this is usually treated in such a way that the vectors are normalized to have length one and afterwards vectors differing by an overall phase e iø are regarded as physically equivalent. In effect then the space of pure states is the complex projective space; as always in this book n = N – 1. The notation used in quantum mechanics differs from what we have used so far. We have denoted vectors in by Z α , while in quantum mechanics they are usually denoted by a ket vector. We can think of the index α as just a label telling us that Z α is a vector, and then these two notations are in fact exactly equivalent.
Abstract This chapter reviews all the rules of quantum mechanics and their mathematical notation. It also considers quantum entanglement, a fundamental resource in quantum information. There are four basic postulates of quantum mechanics, that indicate how to represent physical systems, how to represent observations, how to carry out measurements, and how systems evolve when “not measured”. Before describing the laws of quantum mechanics, their mathematical background is first described, along with a Mach Zehnder interferometer experiment in which the (strange) readings on the detectors can be explained by quantum mechanics. This chapter also discusses the Dirac notation, qubits, Hilbert spaces, projective measurements and operations, unitary operations, eigenvectors and eigenvalues, spectral decomposition, applications of the spectral theorem, Dirac notation shorthands, and mixed states.
Quantum computing (QC) is an exciting new technology that completes the text with an application that relies heavily on the mathematics of linear algebra. There is more to explain than the conversion of common matrix notation into Dirac bra-ket notation. There is also the bizarre world of quantum physics that is inspiring a range of quantum information technologies. Qubits and their entangled superposition states are introduced to show that the “quantum advantage” stems from qubits storing exponentially more information that can be processed at exponentially faster rates. The process of preparing, modifying, and measuring qubit states is explained. Linear algebra skills are exercised by representing QC coding through the application of unitary operators to qubit states. QC algorithms are designed by diagramming quantum circuits, casting those circuits into sequences of operators applied to qubit states, and converting the operators into matrix forms. These are the tools of QC. Several algorithms that demonstrate a range of computational tasks from simple arithmetic through quantum Fourier analysis complete the chapter.
We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.
Abstract In this chapter, we present an overview of linear algebra with the purpose of outlining concepts relevant to quantum computing using the notation common to the field. The chapter consists of two parts. In the first part, we introduce vector spaces. We present the notions of linear dependence, dimensionality, inner product, and distance. We use these concepts to introduce finite-dimensional Hilbert spaces, a key ingredient for quantum information theory. In the second part of the chapter, we introduce linear operators. We describe the main classes of linear operators and show their mutual relations.