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: 1410144
• 负责人: Eric Rowell
• 金额: $ 9万
• 依托单位: Texas A&M University
• 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 2017-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1410144&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.

最近发现的拓扑绝缘体等材料中的拓扑相具有在（量子）计算设备中的潜在用途，这些设备的性能可以超越基于标准微芯片的计算机。最常见的量子计算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，即使不是不可能，也需要具有挑战性的硬件精度才能具有实用价值。基于奇异物质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。该研究项目以数学方式研究物质拓扑相在新计算范式中的应用，这对量子计算具有潜在的重大益处。在该项目中，研究人员研究拓扑相的数学模型，重点关注其在拓扑量子计算中的应用。 物质在两个空间维度中的拓扑相在模块化类别的框架中得到了很好的描述，但在三个空间维度中却知之甚少。 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式。 具体来说，该项目将研究 (3+1) 维拓扑量子场论和环形编织群的表示，以及丰富拓扑序和测量对称性的对称性。 此外，由于局域性和普适性是量子计算的两个理想属性，并在辫群的表示中体现出来，因此该项目还旨在制定表征这些属性何时成立的猜想，并在适当的情况下验证和调整这些猜想，以更好地表征物理和计算方面。 为了将拓扑量子计算机的计算能力与经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计算的复杂性：拓扑不变量。

项目成果

Classification of Super-Modular Categories by Rank

• DOI: 10.1007/s10468-019-09873-9
• 发表时间: 2017-05
• 期刊: Algebras and Representation Theory
• 影响因子: 0.6
• 作者: P. Bruillard;César Galindo;S. Ng;J. Plavnik;E. Rowell;Zhenghan Wang

Modular categories of dimension $p^3m$ with $m$ square-free

• DOI: 10.1090/proc/13776
• 发表时间: 2016-09
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: P. Bruillard;J. Plavnik;E. Rowell