Braiding quantum gates from partition algebras and supersymmetric spin chains

Intriguing connections exist between the topological entanglement of classical objects -- such as knots, links, and braids -- and the entanglement of quantum states. A paradigmatic example is provided by the Borromean rings: a system of three interlocked loops that becomes completely unlinked upon removing any one component. This can be viewed as the topological analogue of the GHZ state of three qubits, a maximally entangled state that collapses to a product state upon measurement of a single qubit. More broadly, it is widely anticipated that topological features of physical systems, particularly those involving anyons, may play a central role in the development of robust, fault-tolerant quantum computers. In this seminar, I will present recent advances exploring these connections. In the first part, I will discuss braiding operators as quantum entangling gates, introducing two novel techniques for generating solutions to the (generalized) Yang–Baxter equation. Such solutions are notoriously difficult to construct and can be interpreted as braiding transformations that produce entangled states from product states of qudits. In the second part, I will review the structure of fusion spaces associated with non-Abelian anyons and demonstrate how these can be effectively emulated using an apparently unrelated system: the zero modes of certain supersymmetric spin chains.
 

Based on papers in collaboration with I. Jana, F. Montorsi, P. Padmanabhan and F. Sugino.