Collaborative Research: Mathematical Foundations of Topological Quantum Computation

最近在拓扑绝缘体等材料中发现的拓扑相在量子计算设备中具有潜在的用途，可以超越基于标准微芯片的计算机。量子计算最常见的模型是量子电路模型，由于系统与周围环境的局部相互作用，它要求硬件上的精度具有挑战性（如果不是不可能的话），以具有实用价值。基于物质的奇异状态的拓扑模型，虽然在数学上更复杂，但对这种相互作用具有内在的容忍度。这个研究项目从数学上研究了物质的拓扑相在量子计算中具有潜在显著优势的新计算范式中的应用。在本项目中，研究人员研究了拓扑相的数学模型，重点研究了它们在拓扑量子计算中的应用。物质在两个空间维度的拓扑相在模范畴的框架中得到了很好的描述，但在三维空间维度中所知相对较少。这个项目的很大一部分致力于在三个空间维度上建立适当的数学模型，并分析相应的计算范式。具体而言，该项目将研究（3+1）维拓扑量子场理论和环形编织群的表示，以及对称丰富的拓扑秩序和测量对称性。此外，由于局部性和普适性是量子计算的两个理想属性，它们在辫群的表示中得到了体现，因此本项目还旨在制定推测，描述这些属性何时成立，并在适当的地方验证和调整这些推测，以更好地描述物理和计算方面。为了比较拓扑量子计算机与经典计算机的计算能力，该项目研究了这种情况下最自然计算的复杂性：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.