Quantum Computation Input ||> Computation ||> Output

In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool, capable of performing tasks which seem intractable for classical computers. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the beneet of being an almost self contained introduction to the exciting eld of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I deene quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor's factorization algorithm and Grover's algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and nite precision. This question cannot be separated from that of quantum complexity, because any realistic model will inevitably be subject to such inaccuracies. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics. 1 Overview Since ancient times, humanity has been seeking tools to help us perform tasks which involve calculations. Such are computing the area of a land, computing the stresses on 1 rods in bridges, or nding the shortest route from one place to another. A common feature of all these tasks is their structure: The computation part of the process is inevitably performed by a dynamical physical system, evolving in time. In this sense, the question of what can be computed, is intermin-gled with the physical question of which systems can be physically realized. If one wants to perform a certain computation task, one should seek the appropriate physical system, such that the evolution in …