Collaborative Research: Mathematical Foundations of Topological Quantum Computation

合作研究：拓扑量子计算的数学基础

• 批准号: 1411212
• 负责人: Zhenghan Wang
• 金额: $ 6万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-12-15 至 2019-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411212&HistoricalAwards=false
• 关键词: Collaborative; Research; Mathematical; Foundations; Topological

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.

最近在诸如拓扑绝缘子等材料中发现的拓扑阶段在（量子）计算设备中具有潜在的用途，该设备可以超过标准的微芯片计算机。量子电路模型最常见的量子计算模型，由于系统与周围环境与周围环境的局部交互，因此，硬件上的精度（即使不是不可能）的准确性必须具有实际价值。基于物质的异国情调状态的拓扑模型虽然数学上更复杂，但具有内在的耐受性。该研究项目从数学上研究了物质的拓扑阶段在量子计算中具有潜在显着益处的新计算范式中的应用。在该项目中，研究人员研究了拓扑阶段的数学模型，重点是他们在拓扑量子计算中的应用。 在两个空间维度中，物质的拓扑阶段在模块类别的框架中很好地描述了，但是在三个空间维度中却鲜为人知。 该项目的很大一部分致力于在三个空间维度上开发适当的数学模型并分析相应的计算范例。 具体而言，该项目将研究（3+1） - 二维拓扑量子场理论和循环辫子组的表示，以及富含对称性的拓扑顺序和测量对称性。 此外，由于局部性和普遍性是量子计算的两个理想特性，这些量子在编织组的表示中表现出来，因此该项目还旨在制定何时表征这些特性的猜想，并在适当的情况下验证和适应这些猜想，以便在适当的情况下更好地表征物理和计算方面。 为了将拓扑量子计算机与古典计算机的计算能力进行比较，该项目研究了这种情况下最自然的计算的复杂性：拓扑不变性。