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

Indiana University, Bloomington IN

Investigators

Abstract

The investigators plan to classify topological quantum field theories (TQFTs) with less than six labels, prove that there are only finitely many TQFTs with a fixed number of labels, and identify the closed images of certain TQFT representations of the mapping class groups of surfaces. For applications to quantum computing the investigators aim to understand how TQFTs would arise from microscopic many-body quantum physics. The proposed program is based on work of Turaev and Moore-Seiberg-Walker which essentially establish a one-one correspondence of TQFTs with modular tensor categories. Topological quantum field theory emerged in the 1980s from the study of three distinct riddles: the relation of the Jones polynomials of knots to 3-dimensional topology, the fractional Quantum Hall effect (FQHE) in condensed matter physics, and the infrared limit of 2-dimensional conformal field theory in string theory. The connection between topological quantum field theory and quantum computing was first explored by Freedman and Kitaev in late 1990s. Their work opened up the possibility of building an inherently fault-tolerant quantum computer---a topological quantum computer. Such a computer would exploit new states of matter closely related to TQFT. Examples of such "topological states of matter" include electron gases confined between the interface of two semi-conductors which exhibit the FQHE. The extreme physical conditions for fractional quantum Hall electron gases make it impractical to build a topological quantum computer from such materials. To discover or fabricate new materials capable of universal quantum computation under practical physical conditions is a goal of the proposed program. The problem of classifying TQFTs (or equivalently topological states of matter) is analogous to the problem of classifying the chemical elements, and can be used to identify appropriate candidates for quantum computing. The proper characterization of materials capable of universal quantum computing, not to speak of their actual fabrication, could open up a new chapter in many-body quantum physics. The possible applications of such new materials are hard to predict, but would definitely not to be limited to quantum computing.