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0-19-857000-7 copy

0-19-857000-7 copy - 1 INTRODUCTION AND BACKGROUND 1.1...

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1 INTRODUCTION AND BACKGROUND 1.1 Overview 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 rela- tionship 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 in- formation 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 correc- tion in detail. 1
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2 INTRODUCTION AND BACKGROUND 1.2 Computers and the Strong Church–Turing Thesis 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. For example, if the problem is to multiply two n bit numbers, a computer might solve this problem using up to 2 n 2 +3 units of time (where the unit of time may be seconds, or the length of time required for the computer to perform a basic step). Of course, the exact amount of resources used by a computer executing an algo- rithm depends on the physical architecture of the computer. A different computer multiplying the same numbers mentioned above might use up to time 4 n 3 + n +5 to execute the same basic algorithm. This fact seems to present a problem if we are interested in studying the complexity of algorithms themselves, abstracted from the details of the machines that might be used to execute them. To avoid
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