Symmetry defects and their application to topological quantum computing

Abstract

We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.

arXiv commentAccepted for publication in Proceedings of the 2016 AMS Special Session on Topological Phases of Matter and Quantum Computation

References in corpus (9)

• Anyons in an exactly solved model and beyond
• Majorana Zero Modes and Topological Quantum Computation
• Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Genons, twist defects, and projective non-Abelian braiding statistics
• Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies
• Degeneracy Implies Non-abelian Statistics
• Topological Qubit Design and Leakage
• The rank of $G$-crossed braided extensions of modular tensor categories