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 2-qubit entangling gates we generate in the present paper, we produce in particular the circulant gate CEG = 1/4 I + I sqrt(3)/4 J - 3/4 J^2 + I sqrt(3)/4 J^3, where J denotes the permutation matrix associated with the cycle (1432) and I denotes the identity matrix.