FRG: Collaborative Research: Topological Quantum Field Theory and its Application to Quantum Computing

点击翻译按钮获取中文摘要

The investigators plan to classify topological quantum field theories(TQFTs) with less than six labels, prove that there are only finitelymany TQFTs with a fixed number of labels, and identify the closed imagesof certain TQFT representations of the mapping class groups of surfaces.For applications to quantum computing the investigators aim tounderstand how TQFTs would arise from microscopic many-body quantumphysics. The proposed program is based on work of Turaev andMoore-Seiberg-Walker whichessentially establish a one-one correspondence of TQFTs with modulartensor categories.Topological quantum field theory emerged in the 1980s from the study ofthree distinct riddles: the relation of the Jones polynomials of knots to3-dimensional topology, the fractional Quantum Hall effect (FQHE) incondensed matter physics, and the infrared limit of 2-dimensionalconformal field theory in string theory. The connection betweentopological quantum field theory and quantum computing was first exploredby Freedman and Kitaev in late 1990s. Their work opened up thepossibility of building an inherently fault-tolerant quantumcomputer---a topological quantum computer. Such a computer wouldexploit new states of matter closely related to TQFT. Examples of such"topological states of matter" include electron gases confined between theinterface of two semi-conductors which exhibit the FQHE. The extremephysical conditions for fractional quantum Hall electron gases make itimpractical to build a topological quantum computer from such materials.To discover or fabricate new materials capable of universal quantumcomputation under practical physical conditions is a goal of the proposedprogram. The problem of classifying TQFTs (or equivalently topologicalstates of matter) is analogous to the problem of classifying thechemical elements, and can be used to identify appropriate candidatesfor quantum computing. The proper characterization of materials capableof universal quantum computing, not to speak of their actual fabrication,could open up a new chapter in many-body quantum physics. Thepossible applications of such new materials are hard to predict,but would definitely not to be limited to quantum computing.