Mathematics of topological quantum computing

Abstract

In topological quantum computing, information is encoded in"knotted" quantum states of topological phases of matter, thus being lockedinto topology to prevent decay. Topological precision has been confirmed inquantum Hall liquids by experiments to an accuracy of 10-10 and harnessedto stabilize quantum memory. In this survey, we discuss the conceptual developmentof this interdisciplinary field at the juncture of mathematics, physics,and computer science. Our focus is on computing and physical motivations,basic mathematical notions and results, open problems and future directionsrelated to and/or inspired by topological quantum computing.