Collaborative Research: Mathematical Foundations of Topological Quantum Computation

最近发现的材料中的拓扑相，如拓扑绝缘体，在(量子)计算设备中具有潜在的用途，这些设备的性能可以超过基于标准微芯片的计算机。最常见的量子计算模型，量子电路模型，由于系统与周围环境的局部相互作用，要求硬件具有挑战性的准确性(如果不是不可能的话)才具有实用价值。基于物质奇异状态的拓扑模型，虽然在数学上更加复杂，但对这种相互作用有内在的容忍度。这个研究项目从数学上研究了物质的拓扑相在新的计算范式中的应用，在量子计算中具有潜在的重大意义。在这个项目中，研究人员研究了拓扑相的数学模型，重点是它们在拓扑量子计算中的应用。物质在两个空间维度上的拓扑相在模块范畴的框架中得到了很好的描述，但在三个空间维度上的研究相对较少。该项目的很大一部分内容是在三维空间中开发适当的数学模型，并分析相应的计算范例。具体地说，该项目将研究(3+1)维拓扑量子场理论和环辫群的表示，对称性丰富了拓扑有序和规范对称性。此外，由于局部性和普适性是量子计算的两个理想性质，表现在辫子群的表示中，因此本项目还旨在制定猜想，以表征这些性质何时成立，并在适当的地方验证和调整这些猜想，以更好地刻画物理和计算方面的特征。为了比较拓扑量子计算机和经典计算机的计算能力，该项目调查了这种环境下最自然的计算的复杂性：拓扑不变量。

Recently discovered topological phases in materials such as topological insulators have potential use in (quantum) computational devices that can out-perform standard microchip based computers. The most commonly encountered model for quantum computation, the quantum circuit model, requires challenging, if not impossible, accuracy on the hardware to be of practical value, due to local interactions of the system with the surrounding environment. The topological model based on exotic states of matter, while mathematically more complicated, has a built-in tolerance for such interactions. This research project studies mathematically the application of topological phases of matter to new computational paradigms with potentially significant benefit in quantum computation.In this project the investigators study mathematical models for topological phases, focusing on their applications to topological quantum computation. Topological phases of matter in two spatial dimensions are well-described in the framework of modular categories, but relatively little is known in three spatial dimensions. A large part of this project is devoted to developing appropriate mathematical models in three spatial dimensions and analyzing the corresponding computational paradigms. Specifically, the project will study (3+1)-dimensional topological quantum field theories and representations of the loop braid group, and symmetry enriched topological order and gauging symmetry. In addition, because locality and universality are two desirable properties for quantum computation that are manifested in the representations of the braid group, this project also aims to formulate conjectures characterizing when these properties hold and to verify and adapt these conjectures where appropriate to better characterize physical and computational aspects. To compare the computational power of topological quantum computers to that of classical computers, the project investigates the complexity of the most natural computation in this setting: topological invariants.