Symmetry Protected and Topological Phases in Quantum Many Body Systems

量子计算的标准方法对量子比特错误的容忍度不高，目前尚不清楚错误率是否可以降低到足够低的水平，以构建通用容错量子计算机。我研究的一个主要重点是探索量子计算的另一种方法，其中量子信息固有地免受错误的影响，并被操纵以执行通用量子逻辑。这项工作的总体框架是利用物质的拓扑绝缘体（TI）和对称保护拓扑（SPT）状态的思想，这些思想最近在凝聚态物理学中引起了强烈的兴趣。在接下来的五年里，我将与实验组密切合作，帮助开发这些新的物质状态，并指导本质上容错的量子算法的实际实现。提出的活动是量子信息和凝聚态物理的独特融合。******也许最著名的TI的例子对应于整数量子霍尔效应，其中电导是用陈氏数来表示的，陈氏数是一种拓扑量。随后，人们意识到拓扑状态可以出现在各种设置中，包括没有对称性破坏的三维（3D）材料（外部磁场破坏量子霍尔效应中的时间反转对称性）。相反，钛相可以直接遵循对称性：SPT相。拓扑有序的标志是表面的局域态；例如，量子霍尔电导是由于这些“边缘模式”。编码到这些模式上的量子信息受到底层对称性/拓扑结构的保护，但仍然可以被外部操纵。对边缘状态的连续测量可以产生通用量子逻辑。******我的研究将考虑最近在实现SPT和TI状态方面取得巨大进展的几个实验系统，例如量子腔和光学晶格中的中性原子以及光子晶体。重点将放在具有自旋轨道耦合的粒子系统上，其基态在一定条件下可以形成SPT相。我将研究理想粒子和相互作用粒子的系统，优先寻找方便的实验方法来检测拓扑秩序和实现量子操作。理论计算将结合分析技术和最先进的数值方法，包括群论、代数图论、矩阵乘积状态和密度矩阵重整化群计算在一维和更一般的张量网络在二维和三维等。这些结果应该有助于引领新型量子技术的发展。

Standard approaches to quantum computing are not very tolerant to qubit errors and it is not yet clear if the error rates can be made sufficiently low to build a universal fault-tolerant quantum computer. A major focus of my research is to explore an alternative approach to quantum computing, where the quantum information is inherently protected from errors and is manipulated to perform universal quantum logic. The general framework of this work is to exploit the ideas of topological insulator (TI) and symmetry-protected topological (SPT) states of matter that have been of intense recent interest in condensed matter physics. Over the course of the next five years, I will work closely with experimental groups to help develop these new states of matter, and to guide the practical implementation of intrinsically fault-tolerant quantum algorithms. The proposed activities are a unique blend of quantum information and condensed matter physics.******Perhaps the best-known example of a TI corresponds to the integer quantum Hall effect, in which the conductance is expressed in terms of the Chern number, a topological quantity. It was subsequently realized that topological states can arise in a variety of settings including three-dimensional (3D) materials with no broken symmetries (the external magnetic field breaks time-reversal symmetry in the quantum Hall effect). Rather, TI phases can follow directly from of the symmetries: SPT phases. A hallmark of the topological order are localized states at the surface; for example, the quantum Hall conductance is due to these 'edge modes.' Quantum information encoded onto these modes is protected by the underlying symmetry / topology, yet remains accessible to external manipulation. Successive measurements of the edge states can yield universal quantum logic.******My research will consider several experimental systems which have recently made huge strides toward the realization of SPT and TI states, such as neutral atoms in quantum cavities and optical lattices, and photonic crystals. Emphasis will be placed on systems of particles with spin-orbit coupling, whose ground states can form SPT phases under certain conditions. I will investigate systems of both ideal and interacting particles, with a priority on finding convenient experimental approaches both for the detection of topological order and for the implementation of quantum operations. The theoretical calculations will employ a combination of analytical techniques and state-of-the-art numerical methods, including group theory, algebraic graph theory, matrix-product states and density matrix renormalization group calculations in 1D and more general tensor networks in 2D and 3D, among others. The results should help lead the way toward to development of novel quantum technologies.