拓扑量子计算的数学理论

Recently discovered topological materials such as topological insulators have potential use in building universal quantum computer. The quantum computation model based on topological phases of matter has a built-in tolerance on the hardware for the interactions of the system with the surrounding environment. Topological phases of matter in two spatial dimensions have been well-described in the framework of modular categories, but relatively little is known in three spatial dimensions. This research project is devoted to developing mathematical models of topological phases of matter in three spatial dimensions: (3+1)-dimensional topological quantum field theories and the motion group of knots with their application to understanding the topology of three and four dimensional manifolds and establishing new quantum computational paradigms in three dimensional materials.

近年来发现的诸如拓扑绝缘体等拓扑材料具有建造通用量子计算机的潜力。基于拓扑物质形态的拓扑量子计算模型在硬件层面对来自环境的干扰具有天然的抵抗。2维的拓扑物质形态已经在模范畴的框架下得到很好的描述，但是我们对3维的拓扑物质形态的数学模型还知之甚少。本项目主要研究3维拓扑物质形态的数学模型：(3+1)维拓扑量子场论和和扭结运动群的表示，并应用于理解3维和4维流形的拓扑以及建立3维材料中量子计算的新模型。

拓扑量子场论是拓扑量子计算的数学基础，提供了在拓扑物质材料上实现量子计算的模型。本项目主要研究了非半单的拓扑量子场论及其诱导的低维流形不变量和子流形运动群表示论，及其在3维拓扑物质和拓扑量子计算中的应用。一个主要结果是解决了关于两种非半单3维流形量子不变量间关系的长期猜想。由于半单的拓扑量子场论被证明不能导出新的4维流形不变量，理解并推广非半单3维流形不变量为进一步研究4维流形的非半单不变量和拓扑打下了基础。另一个成果是使用张量范畴系统地构造了圈辫子群表示。该模型原则上也适用于更一般的纽结运动群。圈辫子群是辫子群的推广和特殊的纽结运动群。类似于辫子群表示被应用于2维拓扑材料中的量子计算模型，圈辫子群等纽结运动群的表示也将应用到3维拓扑材料的研究中。