Making a circulant 2-qubit entangling gate

We present a way to physically realize a circulant 2-qubit entangling gate in the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4. Our approach uses qubit and qutrit ancillas, braids, fusions and interferometric measurements. Our qubit is formed by four anyons of topological charges 1221. Among other 2qubit entangling gates we generate in the present work, we produce in particular the circulant gate CEG = 1 4 I + i √ 3 4 J − 3 4 J + i √ 3 4 J where J denotes the permutation matrix associated with the cycle (1432) and I denotes the identity matrix.