Abstract As discussed in Chap. 10.1007/978-3-030-61601-4_2 , information in classical computers is represented by bits. However, if the bits did not change, then the computer would remain the same forever and would not be very useful! Therefore, it is necessary to change the values of bits depending on what you want the computer to do. For example, if you want a computer to multiply the number 2 and the number 3 together to produce the number 6, then you need to put each of the numbers 2 and 3 into an 8-bit binary representation, and then have a computational operation to multiply the two 8-bit values together to produce 6. The operation of changing bits in a classical computer is performed by classical logic gates.
Abstract As discussed in Chap. 10.1007/978-3-030-61601-4_2 , information in classical computers is represented by bits. However, if the bits did not change, then the computer would remain the same forever and would not be very useful! Therefore, it is necessary to change the values of bits depending on what you want the computer to do. For example, if you want a computer to multiply the number 2 and the number 3 together to produce the number 6, then you need to put each of the numbers 2 and 3 into an 8-bit binary representation, and then have a computational operation to multiply the two 8-bit values together to produce 6. The operation of changing bits in a classical computer is performed by classical logic gates.
Digital computers are based on devices that can take on only two states, one of which is denoted by 0 and the other by 1. By concatenating several 0s and is together, 0-1 combinations can be formed to represent as many different entities as desired. A combination containing a single 0 or 1 is called a bit. In general, n bits can be used to distinguish among 2 n distinct entities and each addition of a bit doubles the number of possible combinations. Computers use strings of bits to represent numbers, letters, punctuation marks, and any other useful pieces of information. In a classical computer, the processing of information is done by logic gate. A logic gate maps the state of its input bits into another state according to a truth table. Quantum computers require quantum logic, something fundamentally different to classical Boolean logic. This difference leads to a greater efficiency of quantum computation over its classical counterpart. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
Abstract Alice and Bob explain the difference between classical (traditional) computing and quantum computing, deploying a gentle introduction to classical binary digits (bits) beginning with a brief history of the development of quantum mechanics and computer architecture. The abstract backbone of classical computing is logic gates, which represent changes to input bits under specific rules. These rules, governed by Boolean, logic can be summarized in truth tables, giving the output values for specific input values. The most common classical gates—NOT, AND, NAND, and XOR gates—are introduced. The vocabulary of classical computing provides the basis for understanding quantum computing.
Computation is a process of performing a large number of simple operations. The main building blocks of classical computers are bits. Each of them can take either the value 0 or 1. A series of gates can change these values by acting on them selectively. This process is structured to fulfil a purpose. For example, if we are interested in adding two numbers, the numbers are encoded into bits and gates are performed such that the final state of the bits reveals the desired answer: a number which is the sum of the initial ones. While it is possible to build algorithms that can perform almost any computation, it is also desirable to find the answer within a reasonable length of time or with a reasonable amount of resources. There is an ever-growing demand in computational power for both scientific and commercial purposes. Indeed, it is surprisingly easy to find an application that can jam even the fastest computer. This fuels a vast effort in the research of information science to increase the speed and processing power of computers. Modern computational models are based on the universal Turing machine (Turing, 1937). This is a theoretical information processing model that employs the elementary gate processes described above. It can efficiently simulate any other device capable of performing an algorithmic process. Since the introduction of the Turing machine, physics has influenced computation in many ways.
The general computational process A suitably programmed quantum computer should act on a number x to produce another number f(x) for some specified function f . Appropriately interpreted, with an accuracy that increases with increasing k , we can treat such numbers as non-negative integers less than 2 k . Each integer is represented in the quantum computer by the corresponding computational-basis state of k Qbits. If we specify the numbers x as n -bit integers and the numbers f(x) as m -bit integers, then we shall need at least n + m Qbits: a set of n -Qbits, called the input register , to represent x , and another set of m -Qbits, called the output register , to represent f(x) . Qbits being a scarce commodity, you might wonder why we need separate registers for input and output. One important reason is that if f(x) assigns the same value to different values of x , as many interesting functions do, then the computation cannot be inverted if its only effect is to transform the contents of a single register from x to f(x) . Having separate registers for input and output is standard practice in the classical theory of reversible computation. Since quantum computers must operate reversibly to perform their magic (except for measurement gates), they are generally designed to operate with both input and output registers. We shall find that this dual-register architecture can also be usefully exploited by a quantum computer in some strikingly nonclassical ways.
Introduction Computation is an operation on symbols. We tend to perceive symbols as abstract entities, such as numbers or letters from a given alphabet. However, symbols are always represented by selected properties of physical objects. The binary string 10011011010011010110101101 may represent an abstract concept, such as the number 40 711 597, but the binary symbols 0 and 1 have also a physical existence of their own. It could be ink on paper (this is most likely to be how you see them when you are reading these words), glowing pixels on a computer screen (this is how I see them now when I am writing these words), or different charges or voltages (this is how my word processor sees them). If symbols are physical objects and if computation is an operation on symbols then computation is a physical process. Thus any computation can be viewed in terms of physical experiments, which produce outputs that depend on initial preparations called inputs. This sentence may sound very innocuous but its consequences are anything but trivial! On the atomic scale matter obeys the rules of quantum mechanics, which are quite different from the classical rules that determine the properties of conventional computers. Today’s advanced lithographic techniques can etch logic gates and wires less than a micrometer across onto the surfaces of silicon chips. Soon they will yield even smaller parts and inevitably reach a point at which logic gates are so small that they are made out of only a handful of atoms. So, if computers are to become smaller in the future, new, quantum technology must replace or supplement what we have now. The point is, however, that quantum technology can offer much more than cramming more and more bits onto silicon and multiplying the clock-speed of microprocessors. It can support an entirely new kind of computation, known as quantum computation, with qualitatively new algorithms based on quantum principles.