The limits of computation

Abstract

The powerful idea of computation has accompanied the development of human civilization, has deeply changed the way we live and work, and has accelerated the advancement of many areas of sciences. In this article, we explore the power and limits of computation from several different perspectives. We will discuss topics from the models of computation and Church-Turing thesis, to the impact of the P versus NP problem and quantum computing on our understanding of the limits of computation. More concretely, we will explore the computability and the halting problem, the efficiency problem of computation, the P versus NP problem. We then move on to the discussion of quantum computation, quantum algorithm for factoring and its implications, quantum simulation and the relation between quantum and classical computations.