Protocol for making a 2-qutrit entangling gate in the Kauffman-Jones version of $SU(2)_4$

The Brylinski couple, in a work [7] dating from 2001, proves that universal single qudit gates together with any 2-qudit entangling gate are sufficient for universal quantum computation. For qubits, there is an independent proof of this fact by [3]. The goal of our present paper is to provide a protocol for producing a 2-qutrit entangling gate in the framework of the Kauffman-Jones version [8] of SU(2) Chern-Simons theory at level 4. Throughout the paper, we assume that the reader is familiar with this theory and a nice reference for it is [8]. We also assume that the reader is familiar with interferometric measurements and we refer the reader to [4], [5], [6]. The anyonic system which we consider is the qutrit formed by four anyons of topological charge 2. The action by the braid group on this qutrit as well as a pair fusion action due to Michael Freedman on the same qutrit are extensively studied in [2] and [9] respectively. The core ideas to produce an entangling gate on the 2-qutrit are to introduce an ancilla which plays the role of a mediator between the left and right qutrits and use it to entangle both qutrits before separating the two qutrits again. Along the way, we need a qubit swap braid that was introduced in [10]. Our protocol uses braids, interferometric measurements, fusions and unfusions of anyons, vacuum pair creation and recovery procedures. These ideas originate in [11].