The simplest quantum system is the physical quantum bit or qubit . The qubit is a two-level quantum system—example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. We do not worry too much about physical implementations in this chapter, but instead focus on the mathematical postulates of the quantum theory and operations that we can perform on qubits. We progress from qubits to a study of physical qudits . Qudits are quantum systems that have d levels and are an important generalization of qubits. Again, we do not discuss physical realizations of qudits. Noise can affect quantum systems, and we must understand methods of modeling noise in the quantum theory because our ultimate aim is to construct schemes for protecting quantum systems against the detrimental effects of noise. In Chapter 1, we remarked on the different types of noise that occur in nature. The first, and perhaps more easily comprehensible type of noise, is that which is due to our lack of information about a given scenario. We observe this type of noise in a casino, with every shuffle of cards or toss of dice. These events are random, and the random variables of probability theory model them because the outcomes are unpredictable. This noise is the same as that in all classical information-processing systems.
The simplest quantum system is the physical quantum bit or qubit . The qubit is a two-level quantum system—example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. We do not worry too much about physical implementations in this chapter, but instead focus on the mathematical postulates of the quantum theory and operations that we can perform on qubits. We progress from qubits to a study of physical qudits . Qudits are quantum systems that have d levels and are an important generalization of qubits. Again, we do not discuss physical realizations of qudits. Noise can affect quantum systems, and we must understand methods of modeling noise in the quantum theory because our ultimate aim is to construct schemes for protecting quantum systems against the detrimental effects of noise. In Chapter 1, we remarked on the different types of noise that occur in nature. The first, and perhaps more easily comprehensible type of noise, is that which is due to our lack of information about a given scenario. We observe this type of noise in a casino, with every shuffle of cards or toss of dice. These events are random, and the random variables of probability theory model them because the outcomes are unpredictable. This noise is the same as that in all classical information-processing systems.
The simplest quantum system is the physical quantum bit or qubit . The qubit is a two-level quantum system—example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. We do not worry too much about physical implementations in this chapter, but instead focus on the mathematical postulates of the quantum theory and operations that we can perform on qubits. From qubits we progress to a study of physical qudits . Qudits are quantum systems that have d levels and are an important generalization of qubits. Again, we do not discuss physical realizations of qudits. Noise can affect quantum systems, and we must understand methods of modeling noise in the quantum theory because our ultimate aim is to construct schemes for protecting quantum systems against the detrimental effects of noise. In Chapter 1, we remarked on the different types of noise that occur in nature. The first, and perhaps more easily comprehensible type of noise, is that which is due to our lack of information about a given scenario. We observe this type of noise in a casino, with every shuffle of cards or toss of dice. These events are random, and the random variables of probability theory model them because the outcomes are unpredictable. This noise is the same as that in all classical information-processing systems. On the other hand, the quantum theory features a fundamentally different type of noise. Quantum noise is inherent in nature and is not due to our lack of information, but is due rather to nature itself. An example of this type of noise is the “Heisenberg noise” that results from the uncertainty principle. If we know the momentum of a given particle from performing a precise measurement of it, then we know absolutely nothing about its position—a measurement of its position gives a random result. Similarly, if we know the rectilinear polarization of a photon by precisely measuring it, then a future measurement of its diagonal polarization will give a random result. It is important to keep the distinction clear between these two types of noise.
Classical information processing is performed using bits , which are just two-state systems, with the two states called 0 and 1. By grouping bits togetherwe can represent arbitrary pieces of information, and by manipulating these bits we can perform arbitrary computations. The corresponding basic element used in quantum information is the quantum bit, or qubit . This is simply a quantum system with two orthonormal basis states, which we shall call |0⧽ and |1⧽. There are many possible physical implementations of a qubit, such as spin states of electrons or atomic nuclei, charge states of quantum dots, atomic energy levels, vibrational states of groups of atoms, polarization states of photons, or paths in an interferometer. At this stage the physical implementation is not important: the idea of a qubit is to abstract the discussion away from physical details. Taking the standard approach of quantum information theory, we shall begin by not worrying too much about the properties of these states, or even what their energies are; we shall simply assume that they are eigenstates of the system's Hamiltonian with known eigenvalues (that is, known energies). This approach allows us to concentrate on the fundamental properties of the system, without considering all the tedious details. We can in principle perform classical information processing on our quantum system by using the two states |0⧽ and |1⧽ as our logical states 0 and 1 and proceeding in the usual fashion, giving rise to the field of reversible computing , which will be explored briefly in Part II. This, however, misses the point.
In general, all forms of information processing can be considered in a quantum mechanical context, including for example communication, channel capacities, and the quantum limits to information extraction. In this chapter we introduce the basic features of quantum information processing. We first present qubits as abstractions of two-level addressable quantum systems, before generalizing this concept to higher-dimensional qudit systems, and continuous variables. We introduce the stabilizer formalism towards the beginning of our discussion as a convenient way to track quantum information. In Section 2.2 we derive the no-cloning theorem, which leads us to cryptography, teleportation, and quantum repeaters. Section 2.3 describes quantum computing with qubits. We start with a brief description of the circuit model, and use it to define cluster states and the one-way model. We study the properties of cluster states and discuss error correction. In the last section we introduce quantum computing over continuous variables. Quantum information Classically, information is carried by ‘bits’, which are physical systems that have two (macroscopic) states denoted by 0 and 1. These bits are described classically, which immediately raises the question of how we should extend this information-theoretic description to the quantum mechanical case. This will lead to the quantum unit of information, the quantum bit or ‘qubit’. 2.1.1 Classical and quantum bits We can define the qubit as a two-dimensional quantum system, which means that the state space of the system is a two-dimensional complex Hilbert space ℋ.
Introduction This chapter gives the physical and mathematical basis of decoherence in quantum systems, with particular emphasis on its importance in quantum information theory and quantum computation. We assume that the reader is already familiar with the basics of quantum mechanics, such as wave functions, the Schrödinger equation, and Hilbert spaces, which are covered in standard textbooks. Nevertheless, all basic concepts will be at least briefly defined and described. A working knowledge of graduate-level linear algebra and calculus is also assumed. In quantum information processing, the term decoherence is often used loosely to describe any kind of noise that can affect a quantum system. This can include sources of noise that seem, qualitatively, to be quite different from each other. These noise sources include: (i) Random driving forces from the environment (e.g., Brownian motion). (ii) Interactions that produce entanglement between the system and the environment. (iii) Statistical imprecision in the experimental controls on the system (e.g., timing errors, frequency fluctuations, etc.). Remarkably, all of these sources of noise can be described by an almost identical mathematical framework, in which pure quantum states can evolve into mixed quantum states, and the quantum effects necessary for information processing – interference, entanglement, and reversible evolution – are disrupted. The chapter begins by examining a highly schematic model of decoherence, involving systems of qubits, but uses this model to present the mathematical formalism used in describing more realistic systems as well.
Classical computing systems are based on electronic circuits represented at a macroscopic level by the electric properties of the circuits. They store information in the form of bits (classical bits) that take values of either 0 or 1, which can be physically realized in the form of electrical signals (for example, low voltage = 0 and high voltage = 1) on electronic circuits. Quantum computer systems differ from classical systems in that their bit level information is directly represented by the interaction of microscopic systems namely, photons, electrons, and atoms, which are monitored and distributed throughout the implementation by quantum theory. The information is stored in the form of qubits (quantum bits) represented as ground states |0| and |1|. Thus, quantum theory represents the weightage of each outcome based on elements composing the physical systems performing each computation of smart systems. In quantum theory, we refer the readers to a timeline depicting the formative years of quantum modules. A growing number of computer scientists and physicists are coming forward, bringing in software packages and quantum programming languages based on classical programming languages. The realization of quantum technology has thus become an international and interdisciplinary effort.
Quantum mechanics is one of the pillars of modern physics, however rather difficult to teach at the introductory level due to the conceptual difficulties and the required advanced mathematics. Nevertheless, attempts to identify relevant features of quantum mechanics and to put forward concepts of how to teach it have been proposed.1–8 Here we present an approach to quantum physics based on the simplest quantum mechanical system—the quantum bit (qubit).1 Like its classical counterpart—the bit—a qubit corresponds to a two-level system, i.e., some system with a physical property that can admit two possible values. While typically a physical system has more than just one property or the property can admit more than just two values, in many situations most degrees of freedom can be considered to be fixed or frozen. Hence a variety of systems can be effectively described as a qubit. For instance, one may consider the spin of an electron or atom, with spin up and spin down as two possible values, and where other properties of the particle such as its mass or its position are fixed. Further examples include the polarization degree of freedom of a photon (horizontal and vertical polarization), two electronic degrees of freedom (i.e., two energy levels) of an atom, or the position of an atom in a double well potential (atom in left or right well). In all cases, only two states are relevant to describe the system.