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

在过去的几年里，围绕着建造一台有用的量子计算机而进行的第二次量子革命取得了戏剧性的进展。物质的拓扑相表现出多体量子纠缠，其重要性已经得到包括2016年诺贝尔物理学奖在内的科学奖的认可。这使得这种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，拓扑量子计算正在第二次量子革命的前沿走向成熟。第二次量子革命的理论基础仍在发展中，但显然，代数及其表示将发挥类似于群论在第一次量子革命中所起的作用。这个重点研究小组旨在阐明拓扑量子计算的理论基础，最终为第二次量子革命奠定理论基础。预计研究成果将指导和加快拓扑量子计算机的建设。一台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面貌。该项目包括研究生和博士后研究员参与跨学科研究。拓扑量子计算的目标是构建一种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阿贝尔任意子的物质的拓扑相。如果一个物理系统在低能量下某些物理量是拓扑不变的，则该物理系统处于拓扑阶段。拓扑特性是非局部的，但可以通过局部几何特性表现出来。拓扑量子计算的成功与否取决于对拓扑相的控制和对其计算能力的理解。这项研究解决了拓扑量子计算的数学、物理和计算方面的问题。这些项目包括超模范畴的分类，模范畴的向量值模形式，模范畴到三维的扩展，共形场理论的模拟，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。这项研究具有潜在的影响，从对顶点算子代数的新理解到有用的量子计算机的开发。一个具体的目标是建立一个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的二维拓扑相的结构理论。

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.