Higher Algebra and Quantum Protocols

该研究项目将整合牛津数学研究所和伯明翰计算机科学学院的专业知识，为几何和量子信息科学中的高维代数计算构建理论和实践工具。通过结构化的研究议程，我们将制定第一个更高类别的组合语言，它适合于计算机实现，并承认自然几何演算，为研究高等表示理论、量子代数和量子计算中的现代问题提供了一个自然的环境。然后，我们将设计和构建这种语言的计算机实现，从而创建一个研究和证明助手，它天生就允许探索和验证高维结构的复杂操作。最后，我们将应用这种计算工具来概括量子代数中的结构，产生量子信息中的新协议，例如非平面状态的量子纠错。这项研究工作将组织成四个方面-基础、计算机实现、图形微积分和量子协议--通过联合两个机构和学科的研究人员以及通过积极参与更广泛的科学界，将实现雄心勃勃的综合研究议程。

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.