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

研究人员计划对少于6个标号的拓扑量子场论(TQFT)进行分类，证明具有固定标号的TQFT只有有限多个，并确定曲面映射类群的某些TQFT表示的闭像。为了将其应用于量子计算，研究人员的目的是了解TQFT如何从微观多体量子物理中产生。该程序基于Turaev和Moore-Seiberg-Walker的工作，实质上建立了TQFT与模张量范畴的一一对应。20世纪80年代，拓扑量子场论诞生于对三个不同谜团的研究中：纽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。拓扑量子场论和量子计算之间的联系最早是由弗里德曼和基塔耶夫在20世纪90年代末提出的。他们的工作开启了建造一台具有内在容错能力的量子计算机的可能性-一种拓扑量子计算机。这样的计算机将开发出与TQFT密切相关的物质的新状态。这种“物质的拓扑态”的例子包括限制在两个半导体界面间的电子气，它们表现出FQHE。分数量子霍尔电子气的极端物理条件使得利用这种材料构建拓扑量子计算机是非常现实的。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提出的计划的目标。对TQFT(或相当于物质的拓扑态)进行分类的问题类似于对化学元素进行分类的问题，可以用来确定适合进行量子计算的候选者。对能够用于普遍量子计算的材料进行适当的表征，更不用说它们的实际制造了，可能会开启多体量子物理学的新篇章。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.