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.
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 Part II of this book we show how computations can be implemented using quantum systems. As we will see, the differences between bits and qubits, briefly outlined in Part I, lead to some important consequences for quantum computing. We begin with a brief introduction to the fundamental principles underlying quantum computing: specific implementations will be considered in later chapters. Reversible computing While quantum computation in its modern form is still a relatively young discipline, researchers have been interested in the relationship between quantum mechanics and computing for a long time. Early workers were not interested in the ideas of quantum parallelism , which will be explored in the next section, but rather in the question of whether explicitly quantum mechanical systems could be used to implement classical computations. In addition to its intrinsic interest, there are two technological reasons why this might be considered an important question. The first reason is a direct consequence of Moore's laws. After the development of integrated circuits, computing technology began its headlong dash down the twin roads of ever-faster and ever-smaller devices. These two phenomena are closely related: as computing devices must communicate within themselves, and as the speed of information transfer cannot exceed the speed of light, faster computers must indeed be smaller. There is, however, a limit to this process, defined by the atomic scale: once the size of individual transistors becomes comparable with that of atoms, the old-fashioned approach of micro-electronics becomes completely untenable.
From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician. Sir James Jeans, Mysterious Universe 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. Reader Tip . Discussion of the actual physical implementation of qubits and quantum gates is dealt with in Chapter 11. BITS AND QUBITS What is a bit ? Definition 5.1.1 A bit is a unit of information describing a two-dimensional classical system . There are many examples of bits: ▪ A bit is electricity traveling through a circuit or not (or high and low). ▪ A bit is a way of denoting “true” or “false.” […]
We have seen how information can be encoded onto a quantum system by selecting the state in which it is prepared. Retrieving the information is achieved by performing a measurement, and the optimal measurement in any given situation is usually a generalized measurement. In between preparation and measurement, the information resides in the quantum state of the system, which evolves in a manner determined by the Hamiltonian. The associated unitary transformation may usefully be viewed as quantum information processing; if we can engineer an appropriate Hamiltonian then we can use the quantum evolution to assist in performing computational tasks. Our objective in quantum information processing is to implement a desired unitary transformation. Typically this will mean coupling together a number, perhaps a large number, of qubits and thereby generating highly entangled states. It is fortunate, although by no means obvious, that we can realize any desired multiqubit unitary transformation as a product of a small selection of simple transformations and, moreover, that each of these need only act on a single qubit or on a pair of qubits. The situation is reminiscent of digital electronics, in which logic operations are decomposed into actions on a small number of bits. If we can realize and control a very large number of such operations in a single device then we have a computer. Similar control of a large number of qubits will constitute a quantum computer. It is the revolutionary potential of quantum computers, more than any other single factor, that has fuelled the recent explosion of interest in our subject. We shall examine the remarkable properties of quantum computers in the next chapter. In digital electronics, we represent bit values by voltages: the logical value 1 is a high voltage (typically +5 V) and 0 is the ground voltage (0 V). The voltage bits are coupled and manipulated by transistor-based devices, or gates. The simplest gates act on only one bit or combine two bits to generate a single new bit, the value of which is determined by the two input bits. For a single bit, with value 0 or 1, the only possible operations are the identity (which does not require a gate) and the bit flip.
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.
In quantum computers a unitary transformation is applied to a given initial state of a set of qubits through several quantum gates. Tests were conducted using Grover's algorithm and implemented in a variety of different ways on the publically available IBM quantum computers. Essentially, quantum computers manipulate quantum states instead of classical bits. Quantum information can be represented by the elementary units called quantum bits abbreviated as qubits or qbits. This chapter discusses the fundamental properties of quantum systems that are relevant to information processing. Quantum computation needs something much more than that quantum parallelism to be useful. This is achieved in Deutsch's algorithm which combines quantum parallelism with interference. One can simulate a classical circuit using a quantum circuit. That is, it is possible to perform classical computations on quantum computers. But the advantage of quantum computing is that powerful functions may be computed by making use of quantum parallelism, the fundamental feature of many quantum algorithms.
This Spotlight aims to provide a general introduction to linear and nonlinear optical components that are frequently used to implement the protocols for quantum computation and communication. The role of each optical element is described briefly, and it is shown that these optical elements can be combined to constitute quantum circuits for performing various quantum computing and communication tasks. Present challenges and future scopes are also mentioned briefly. Specifically, after introducing the qubit, it is stated that a quantum gate is a device that transforms a quantum state into another quantum state. Thus, most of the optical components (e.g., a beamsplitter, a wave plate, a mirror, a polarizing beamsplitter) can be viewed as quantum gates because each of these optical elements can transform a quantum state into another quantum state. This fact is the basis for constructing useful quantum circuits using optical elements. Finally, a few proposals for optical realizations of schemes for quantum communication and computation are described.
In classical computation and communication, we use classical gates and bits. Similarly, in quantum computation and communication, we use their quantum analogs, i.e., quantum gates and qubit(s) [quantum bit(s)]. We have already mentioned that a bit can be realized in various ways. In the same way, a qubit can also be realized in various ways, and accordingly, corresponding quantum gates that would operate on qubits can also be realized in various ways. Here, we focus on a specific implementation: optical implementation, where qubits are realized using photons and quantum gates, and circuits are realized using various optical components. Specific merits of this realization lie in the following facts: (i) in quantum communication we need to transmit qubits from one location to the other, and transmission of photons is easier than any other realization of qubits; (ii) the effect of the environment (e.g., external electric and magnetic fields) on photons is less compared to any other implementations of qubits; and (iii) integrated optics are already a well-developed field. Keeping these points in mind, this Spotlight describes basic optical elements and how to use those elements to implement various schemes of quantum computation and communication. We will mostly describe realizations using bulk optics. However, in most cases, it is not difficult to transform these bulk-optics-based realizations into integrated optics. To begin with, in Section 1.1, we formally define a qubit and describe two alternative ways of realizing a qubit using a photon.