Urs Schreiber

September, 2022

Title: Knots for quantum computation from defect branes

Abstract: Already 25 years ago [1], Kitaev proposed that intrinsically fault-tolerant quantum computation should be possible by knotting the worldlines of defect points in effectively 2-dimensional quantum materials akin to graphene. Meanwhile, there have been striking advances (a) in the theoretical understanding of such quantum materials, in terms of topological K-theory; and (b) in the practical construction of toy quantum computers – but the “topological quantum gates” proposed by Kitaev have remained somewhat elusive, both practically but also theoretically. Recently we have shown [2] that a previously neglected sector of topological K-theory in its fully-fledged “twisted & equivariant & differential” refinement does reflect those topological quantum gates that are thought to be practically realizable – namely the “su(2)-anyon braid gates”. This insight was drawn from the study of “defect branes” in string theory and points to a non-perturbative enhancement of what is known as “holographic” condensed matter theory. I will give a gentle motivation and exposition of these results. Talk notes will be available at: ncatlab.org/schreiber/show/Erice+2022

This is joint work with Hisham Sati.

[1] Kitaev, A. 1997 Faul-tolerant quantum computation by anyons. Annals of Physics, 303, 2-30. arXiv:quant-ph/9707021
[2] Sati, H. & Schreiber, U. 2022 Topological Quantum Programming in TED-K, PlanQC 2022, 33.

Keywords: worldlines, topological K-theory, anyon braid.