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

研究人员计划用少于6个标签对拓扑量子场论（TQFT）进行分类，证明只有有限数量的TQFT具有固定数量的标签，并识别映射类群的某些TQFT表示的封闭图像。对于量子计算的应用，研究人员的目标是了解tqft如何从微观多体量子物理学中产生。提出的方案是基于Turaev和moore - seiberg - walker的工作，他们基本上建立了具有模张量类别的TQFTs的一对一对应关系。拓扑量子场论出现于20世纪80年代，起源于对三个不同谜题的研究：结的琼斯多项式与三维拓扑的关系，凝聚态物理中的分数量子霍尔效应（FQHE），以及弦理论中二维共形场论的红外极限。弗里德曼和基塔耶夫在20世纪90年代末首次探索了拓扑量子场论与量子计算之间的联系。他们的工作开启了构建固有容错量子计算机——拓扑量子计算机的可能性。这样的计算机将利用与TQFT密切相关的物质的新状态。这种“物质的拓扑状态”的例子包括限制在两个半导体界面之间的电子气体，它们表现出FQHE。分数量子霍尔电子气体的极端物理条件使得用这种材料构建拓扑量子计算机变得不切实际。发现或制造能够在实际物理条件下进行通用量子计算的新材料是拟议计划的目标。对tqft（或物质的等效拓扑状态）进行分类的问题类似于对化学元素进行分类的问题，可用于确定量子计算的适当候选者。对具有通用量子计算能力的材料的适当描述，更不用说它们的实际制造，可能会为多体量子物理学开辟新的篇章。这种新材料的可能应用很难预测，但肯定不会局限于量子计算。

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.