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Birkhäuser Basel eBooks

Quantum Hardware

Yorick Hardy, Willi-Hans Steeb

Published Jan 1, 2001• 0 citations
Publisher

Abstract

Computation is ultimately a physical process. In practice, the range of physically realizable devices determines what is computable and the resources, such as computer time, required to solve a given problem. Computing machines can exploit a variety of physical processes and structures to provide distinct trade-offs in resource requirements. An example is the development of parallel computers with their trade-off of overall computation time against the number of processors employed. 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.

DOI
10.1007/978-3-0348-8366-5_24

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Computation is ultimately a physical process. In practice, the range of physically realizable devices determines what is computable and the resources, such as computer time, required to solve a given problem. Computing machines can exploit a variety of physical processes and structures to provide distinct trade-offs in resource requirements. An example is the development of parallel computers with their trade-off of overall computation time against the number of processors employed. 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.
A general quantum machine is characterized to capture all the possible physical structures that may be employed for effective or efficient computation. As an illustration, a specific machine is described for the permanent evaluation problem. The contrast between quantum and classical parallelism is also discussed, as is the effect of quantum error correction against decoherence and loss. 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.
A computer is a physical device that helps us process information by executing algorithms. An algorithm is a well-defined procedure, with finite description, for realizing an information-processing task. An information-processing task can always be translated into a physical task. When designing complex algorithms and protocols for various information-processing tasks, it is very helpful, perhaps essential, to work with some idealized computing model. However, when studying the true limitations of a computing device, especially for some practical reason, it is important not to forget the relationship between computing and physics. Real computing devices are embodied in a larger and often richer physical reality than is represented by the idealized computing model. Quantum information processing is the result of using the physical reality that quantum theory tells us about for the purposes of performing tasks that were previously thought impossible or infeasible. Devices that perform quantum information processing are known as quantum computers. In this book we examine how quantum computers can be used to solve certain problems more efficiently than can be done with classical computers, and also how this can be done reliably even when there is a possibility for errors to occur. In this first chapter we present some fundamental notions of computation theory and quantum physics that will form the basis for much of what follows. After this brief introduction, we will review the necessary tools from linear algebra in Chapter 2, and detail the framework of quantum mechanics, as relevant to our model of quantum computation, in Chapter 3. In the remainder of the book we examine quantum teleportation, quantum algorithms and quantum error correction in detail. We are often interested in the amount of resources used by a computer to solve a problem, and we refer to this as the complexity of the computation. An important resource for a computer is time. Another resource is space, which refers to the amount of memory used by the computer in performing the computation. We measure the amount of a resource used in a computation for solving a given problem as a function of the length of the input of an instance of that problem.
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.
We are now ready to assemble quantum subroutines into quantum algorithms. A reasonable first question is: Are there problems which can be solved efficiently on a quantum computer? In his analysis of a universal quantum computer [25], Deutsch illustrated quantum parallelism by constructing a procedure to compute the parity f (0) ⊕ f (1) of a function on one bit. In a generalization of this context, Deutsch and Jozsa [27] defined a quantum algorithm that could solve a problem more efficiently than is possible on a classical computer. This problem helped motivate the search for algorithms that could solve “real” problems, and Simon [67] provided an example that inspired Shor’s subsequent work on the factoring problem and on the discrete logarithm problem. 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.
As already emphasized in Chap. 4 information is physical and the limits to information processing tasks are ultimately set by the underlying physical laws. For instance, the standing models of computation are based on the physics of deterministic and/or stochastic classical processes; instead, quantum computation theory [84, 90, 156, 198, 266] studies the new possibilities offered by a model of computation based on quantum mechanical laws. The birth of such a theory finds a technological motivation in the high pace at which chip miniaturization proceeds.