A non-semisimple version of the Kitaev model

Abstract: In 1997, Alexei Kitaev proposed a foundational model for fault-tolerant quantum computation based on complex semisimple Hopf algebras. Its key feature is a topologically invariant code space which is constructed using combinatorial data encoded by a graph embedded into a closed oriented surface. This ensures robustness against a wide range of errors. Beyond applications in quantum computing, the model has remarkable connections with combinatorics, Hopf algebra representation theory, homological algebra, and topological quantum field theories. In this talk, based on joint work with U.\ Krähmer, we present a generalisation of the Kitaev model to arbitrary finite-dimensional Hopf algebras. Two challenges prevent a straightforward approach. First, the extended Hilbert space, a Yetter--Drinfeld module whose invariant submodule is the code space, relies on an involutive antipode---a condition equivalent to the underlying Hopf algebra being semisimple. Second, topological invariance is proven using projectors assembled from (co)integrals. Since we do not have these tools at our disposal, we follow a new approach, inspired by homological considerations. We introduce involutive Hopf bimodules, which are related to coefficients of Hopf cyclic cohomology and allow us to form appropriate, Yetter–Drinfeld valued, variants of extend Hilbert spaces. Instead of considering invariant submodules, the analoga of the code spaces arise as bitensor products---combinations of cotensor and tensor products. Our proof of their topological invariance relies on a notion of excision and uses actions of a group related to mapping class groups. Towards computing bitensor products, we discuss induction-restriction type identities, which are particularly useful for eg. small quantum groups.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home