FRG: cQIS: Collaborative Research: Mathematical Foundations of Topological Quantum Computation and Its Applications

FRG：cQIS：协作研究：拓扑量子计算的数学基础及其应用

基本信息

批准号：
1664351
负责人：
Zhenghan Wang
金额：
$ 36.81万
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：
2017
资助国家：
美国
起止时间：
2017-12-01 至 2023-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664351&HistoricalAwards=false
关键词：
FRG cQIS Collaborative Research Mathematical

项目摘要

A second quantum revolution in and around the construction of a useful quantum computer has been advancing dramatically in the last few years. Topological phases of matter, the importance of which has been recognized by scientific awards that include the 2016 Nobel prize in physics, exhibit many-body quantum entanglement. This makes such materials prime candidates for use in a quantum computer. Topological quantum computation is maturing at the forefront of the second quantum revolution as a primary application of topological phases of matter. The theoretical foundation for the second quantum revolution remains under development, but it appears clear that algebras and their representations will play a role analogous to that played by group theory in the first quantum revolution. This focused research group aims to formulate the theoretical foundations of topological quantum computation, leading to an eventual theoretical foundation for the second quantum revolution. It is anticipated that the results of the research will guide and accelerate the construction of a topological quantum computer. A working topological quantum computer will fundamentally transform the landscape of information science and technology. The project includes participation by graduate students and postdoctoral associates in the interdisciplinary research.The goal of topological quantum computation is the construction of a useful quantum computer based on braiding anyons. The hardware of an anyonic quantum computer will be a topological phase of matter that harbors non-abelian anyons. A physical system is in a topological phase if at low energies some physical quantities are topologically invariant. Topological properties are non-local, yet can manifest themselves through local geometric properties. The success of topological quantum computation hinges on controlling topological phases and understanding their computational power. This research addresses the mathematical, physical, and computational aspects of topological quantum computation. The projects include classification of super-modular categories, vector-valued modular forms for modular categories, extension of modular categories to three dimensions, simulation of conformal field theories, topological quantum computation with gapped boundaries and symmetry defects, and universality of topological computing models. The research has potential impacts ranging from new understanding of vertex operator algebras to the development of useful quantum computers. One specific goal is a structure theory of modular categories analogous to that of finite groups. Such a theory would lead to a structure theory of two-dimensional topological phases of matter.

过去几年，围绕有用量子计算机的构建的第二次量子革命取得了巨大进展。物质的拓扑相表现出多体量子纠缠，其重要性已得到包括 2016 年诺贝尔物理学奖在内的科学奖项的认可。这使得此类材料成为量子计算机中使用的主要候选材料。作为物质拓扑相的主要应用，拓扑量子计算在第二次量子革命的前沿日趋成熟。第二次量子革命的理论基础仍在发展中，但很明显，代数及其表示形式将发挥与第一次量子革命中群论类似的作用。这个重点研究小组的目标是建立拓扑量子计算的理论基础，为第二次量子革命奠定最终的理论基础。预计该研究成果将指导和加速拓扑量子计算机的建设。工作的拓扑量子计算机将从根本上改变信息科学和技术的格局。该项目包括研究生和博士后参与的跨学科研究。拓扑量子计算的目标是构建一台基于编织任意子的有用的量子计算机。任意子量子计算机的硬件将是包含非交换任意子的物质拓扑相。如果在低能量下某些物理量是拓扑不变的，则物理系统处于拓扑相。拓扑性质是非局部的，但可以通过局部几何性质表现出来。拓扑量子计算的成功取决于控制拓扑相并了解其计算能力。这项研究涉及拓扑量子计算的数学、物理和计算方面。这些项目包括超模范畴的分类、模范畴的向量值模形式、模范畴向三维的扩展、共形场论的模拟、具有间隙边界和对称缺陷的拓扑量子计算以及拓扑计算模型的普适性。该研究具有从对顶点算子代数的新理解到有用量子计算机的开发等潜在影响。一个具体的目标是建立类似于有限群的模范畴的结构理论。这样的理论将导致物质二维拓扑相的结构理论。