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。

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.