Recently, a pair of experiments (Lu et al 2009 Phys. Rev. Lett. 102 030502; Pachos et al 2009 New J. Phys. 11 083010) demonstrated the simulation of Abelian anyons in a spin network of single photons. The experiments were based on the Abelian discrete gauge theory spin lattice model of Kitaev (Kitaev 2003 Ann. Phys., NY 303 2). Here, we propose an experiment using linear optics and single photons to simulate non-Abelian anyons. The scheme makes use of joint qutrit-qubit encoding of the spins, and the resources required are three pairs of parametric downconverted photons and 14 beam splitters.