Higher Algebra and Quantum Protocols

Lead Research Organisation: University of Birmingham
Department Name: School of Computer Science


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.


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M. Anel (2020) A Generalized Blakers-Massey Theory in Journal of Topology

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Reutter DJ (2019) Shaded tangles for the design and verification of quantum circuits. in Proceedings. Mathematical, physical, and engineering sciences

Title Catt.io 
Description Software tool for working with syntactic expressions in higher category theory 
Type Of Technology New/Improved Technique/Technology 
Year Produced 2019 
Impact Accelerated research progress in this area of mathematics. 
URL https://arxiv.org/abs/1703.09050
Title homotopy.io 
Description Research tool for higher category theory 
Type Of Technology Webtool/Application 
Year Produced 2019 
Open Source License? Yes  
Impact Increased impact of scientific results in the community 
URL http://homotopy.io
Description Qubit Zone public engagement workshop 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Public/other audiences
Results and Impact Public workshop that allows the general public to interact with quantum computer simulations. Have delivered it over 20 times to around 20-30 participants per event. Hugely successful ongoing outreach programme.
Year(s) Of Engagement Activity 2017,2018,2019,2020