Symmetry Protected and Topological Phases in Quantum Many Body Systems

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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.