Collaborative Research: FET: Small: Topological quantum computing beyond anyons

量子计算最近引起了人们的广泛关注，学术界、工业界和政府的投资迅速增加，用于建造大规模有用的量子计算机。实现这一目标的最佳方法之一是拓扑量子计算（TQC）。TQC的硬件由物质的拓扑相提供-这是2016年诺贝尔物理学奖的主题。 TQC目前依赖于拓扑量子介质中的准粒子激发或任意子来执行计算。任意子的移动性引入了局部噪声，这是扩展量子计算的主要障碍。任意子也很难实现和操纵-它们以二维相实现，而三维（3D）材料更可用和实用。随着任意子基本数学理论的建立，以及相关的既有前景又有挑战性的实验的广泛开展，该项目试图将TQC的范围扩展到二维任意子之外。目标是不仅提供新的方法，TQC超越任何子，但也加强拓扑物理，更高的范畴理论和拓扑之间的联系。该项目的重点是量子纠错，计算普遍性和超越任何子的TQC的数学基础。该项目系统地研究了三个相关方面：分形相位，三维拓扑相位中的环激发和对称缺陷。首先，研究人员将探索分形相的纠错特性和数学基础，这些分形相形成了一类奇异的晶格模型，无法用传统的拓扑量子场论描述。其次，该项目将开发计算物质的3D拓扑相中的环激发的交换统计的方法。 利用表示论和拓扑学中的新技术来计算与3D拓扑场论相关联的运动群表示。第三，研究拓扑相位与对称缺陷之间的相互作用。像任意子一样，缺陷也提供稳定的基态简并和非阿贝尔统计，因此可以用来执行比单独的任意子更强大的量子计算。研究人员对阿贝尔任意子产生的非阿贝尔缺陷特别感兴趣，并试图研究它们是否可以在现实物理条件下普遍用于量子计算。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

Quantum computing has drawn a vast amount of attention recently, with investment from across academia, industries, and governments rapidly increasing for the construction of a large-scale useful quantum computer. Among the best approaches to this goal is topological quantum computing (TQC). The hardware for TQC is provided by topological phases of matter -- the subject of the 2016 Physics Nobel prize. TQC currently relies on quasi-particle excitations or anyons in topological quantum media to perform computing. The mobility of anyons introduces local noise, a major impediment to scaling quantum computing. Anyons are also difficult to realize and manipulate - they are realized in two-dimensional phases, while three-dimensional (3D) materials are more available and practical. With the mathematical theories underlying anyons mostly established and the relevant both promising and challenging experiments being carried out extensively, the project seeks to expanding the scope of TQC beyond just two-dimensional anyons. The goal is to not only provide new approaches to TQC beyond anyons, but also strengthen the connection between topological physics, higher category theory, and topology. The key foci of the project are on quantum error correction, computational universality, and mathematical foundations of TQC beyond anyons. The project systemically investigates three related aspects: fracton phases, loop excitations in 3D topological phases, and symmetry defects. Firstly, the investigators will explore error correction properties and mathematical foundations of fracton phases, which form an exotic class of lattice models that cannot be described by conventional topological quantum field theories. Secondly, the project will develop methods to compute the exchange statistics of loop excitations in 3D topological phases of matter. New techniques in representation theory and topology are utilized to compute motion group representations associated with 3D topological field theories. Thirdly, the project will study the interaction between topological phases and symmetry defects. The defects, like anyons, also provide stable ground state degeneracy and non-Abelian statistics, and hence can be used to perform quantum computing more powerfully than anyons alone. The investigators are especially interested in non-Abelian defects arising from Abelian anyons and seek to examine if they can be made universal for quantum computing under realistic physical conditions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.