Higher Algebra and Quantum Protocols

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This research project will integrate expertise in the Oxford Mathematical Institute and the Birmingham School of Computer Science to build theoretical and practical tools for higher-dimensional algebraic computations in geometry and quantum information science. Through a structured research agenda, we will formulate the first combinatorial language of higher categories that is amenable to computer implementation and admits a natural geometric calculus, providing a natural setting to investigate modern questions in higher representation theory, quantum algebra, and quantum computation. We will then design and build a computer implementation of this language and thereby create a research and proof assistant that natively allows the exploration and verification of complex manipulations of high-dimensional structures. Finally we will apply this computational tool to generalise structures in quantum algebra, yielding novel protocols in quantum information, such as quantum error correction of nonplanar states.This research effort will be organised into four strands---Foundations, Computer Implementation, Graphical Calculus, and Quantum Protocols---and the ambitious, integrated research agenda will be achieved by uniting researchers across the two institutions and disciplines, and by actively engaging with the broader scientific community.