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.

在过去的几年中，有用的量子计算机的第二次量子革命一直在巨大发展。物质的拓扑阶段，包括2016年诺贝尔物理学奖在内的科学奖项所认可的，其重要性表现出多体量子纠缠。这使此类材料是用于量子计算机中的主要候选者。拓扑量子计算在第二量子革命的最前沿，作为物质拓扑阶段的主要应用。第二个量子革命的理论基础仍在开发中，但很明显，代数及其代表将发挥类似于第一次量子革命中群体理论的作用。这个重点研究小组旨在制定拓扑量子计算的理论基础，从而导致第二次量子革命的理论基础。预计研究结果将指导和加速拓扑量子计算机的构建。工作拓扑量子计算机将从根本上改变信息科学和技术的景观。该项目包括研究生和博士后同事参与跨学科研究。拓扑量子计算的目的是建造基于编织者的有用量子计算机。 Anyonic量子计算机的硬件将是具有非亚伯人的物质的拓扑阶段。如果在低能量下，某些物理量在拓扑上是不变的，则物理系统处于拓扑阶段。拓扑特性是非本地的，但可以通过局部几何特性表现出来。拓扑量子计算的成功取决于控制拓扑阶段并了解其计算能力。这项研究涉及拓扑量子计算的数学，物理和计算方面。这些项目包括对模块类别类别的超模块化类别的分类，矢量值模块化形式，模块类别的扩展到三个维度，保形场理论的仿真，拓扑量子计算，具有间隙边界和对称性缺陷，以及拓扑计算模型的普遍性。这项研究具有潜在的影响，从对顶点操作员代数的新理解到有用的量子计算机的开发。一个具体的目标是模块类别类别的结构理论类似于有限群体。这种理论将导致物质二维拓扑阶段的结构理论。