Supervisor: Professor Shahn Majid
Project description:
Quantum computing hardware is subject to noise and critical to further progress is the search for methods to build in quantum fault tolerance. The project will use the methods of quantum algebra and quantum geometry to develop and study new processes for fault tolerant quantum computing and quantum error correction.
The mathematics for this in the case of topologically fault tolerant `surface codes’ is closely related on the one hand to quantum groups, Hopf algebras, braided categories[1] and quantum geometry[2] on the mathematics side and to condensed matter and topological quantum field theory on physics side [3]. Quantum groups emerged in the 1980s as a generalised notion of `quantum symmetry’ in certain systems. They are also typically the symmetries of noncommutative or `quantum’ spaces, in which geometry is generalised to allow noncommutative or `quantum’ coordinates.
The project will draw on this background to understand current ideas for surface codes, particularly those involving nonabelian anyons such as in the Kitaev model based on the quantum group D(G) associated to a nonabelian finite group G, extending work such as [4]. We will then consider generalisations to D(H) where H is a finite-dimensional Hopf algebra or quantum group and develop a suitably general ZX-calculus needed to realise quantum computing gates in this context. There are also links to quantum information and quantum gravity.
The project will be supported by involvement with the quantum computing company Quantinuum.
[1] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press (2000) [2] E.J. Beggs, S. Majid, Quantum Riemannian Geometry, Grundlehren vol. 355, Springer (2020)[3] A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303 (2003) 3–20 [4] A. Cowtan, S. Majid, Quantum double aspects of surface code models, J. Math. Phys. 63 (2022) 042202
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