Collaborative Research: Mathematical Foundations of Topological Quantum Computation

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Recently discovered topological phases in materials such as topological insulators have potential use in (quantum) computational devices that can out-perform standard microchip based computers. The most commonly encountered model for quantum computation, the quantum circuit model, requires challenging, if not impossible, accuracy on the hardware to be of practical value, due to local interactions of the system with the surrounding environment. The topological model based on exotic states of matter, while mathematically more complicated, has a built-in tolerance for such interactions. This research project studies mathematically the application of topological phases of matter to new computational paradigms with potentially significant benefit in quantum computation.In this project the investigators study mathematical models for topological phases, focusing on their applications to topological quantum computation. Topological phases of matter in two spatial dimensions are well-described in the framework of modular categories, but relatively little is known in three spatial dimensions. A large part of this project is devoted to developing appropriate mathematical models in three spatial dimensions and analyzing the corresponding computational paradigms. Specifically, the project will study (3+1)-dimensional topological quantum field theories and representations of the loop braid group, and symmetry enriched topological order and gauging symmetry. In addition, because locality and universality are two desirable properties for quantum computation that are manifested in the representations of the braid group, this project also aims to formulate conjectures characterizing when these properties hold and to verify and adapt these conjectures where appropriate to better characterize physical and computational aspects. To compare the computational power of topological quantum computers to that of classical computers, the project investigates the complexity of the most natural computation in this setting: topological invariants.