Quantum computing has drawn a vast amount of attention recently, with investment from across academia, industries, and governments rapidly increasing for the construction of a large-scale useful quantum computer. Among the best approaches to this goal is topological quantum computing (TQC). The hardware for TQC is provided by topological phases of matter -- the subject of the 2016 Physics Nobel prize. TQC currently relies on quasi-particle excitations or anyons in topological quantum media to perform computing. The mobility of anyons introduces local noise, a major impediment to scaling quantum computing. Anyons are also difficult to realize and manipulate - they are realized in two-dimensional phases, while three-dimensional (3D) materials are more available and practical. With the mathematical theories underlying anyons mostly established and the relevant both promising and challenging experiments being carried out extensively, the project seeks to expanding the scope of TQC beyond just two-dimensional anyons. The goal is to not only provide new approaches to TQC beyond anyons, but also strengthen the connection between topological physics, higher category theory, and topology.

The key foci of the project are on quantum error correction, computational universality, and mathematical foundations of TQC beyond anyons. The project systemically investigates three related aspects: fracton phases, loop excitations in 3D topological phases, and symmetry defects. Firstly, the investigators will explore error correction properties and mathematical foundations of fracton phases, which form an exotic class of lattice models that cannot be described by conventional topological quantum field theories. Secondly, the project will develop methods to compute the exchange statistics of loop excitations in 3D topological phases of matter. New techniques in representation theory and topology are utilized to compute motion group representations associated with 3D topological field theories. Thirdly, the project will study the interaction between topological phases and symmetry defects. The defects, like anyons, also provide stable ground state degeneracy and non-Abelian statistics, and hence can be used to perform quantum computing more powerfully than anyons alone. The investigators are especially interested in non-Abelian defects arising from Abelian anyons and seek to examine if they can be made universal for quantum computing under realistic physical conditions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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Purdue University
West Lafayette
United States
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