  Engineering complex non-Abelian anyon models with simple physical systems is
crucial for topological quantum computation. Unfortunately, the simplest
systems are typically restricted to Majorana zero modes (Ising anyons). Here we
go beyond this barrier, showing that the $\mathbb{Z}_4$ parafermion model of
non-Abelian anyons can be realized on a qubit lattice. Our system additionally
contains the Abelian $D(\mathbb{Z}_4)$ anyons as low-energetic excitations. We
show that braiding of these parafermions with each other and with the
$D(\mathbb{Z}_4)$ anyons allows the entire $d=4$ Clifford group to be
generated. The error correction problem for our model is also studied in
detail, guaranteeing fault-tolerance of the topological operations. Crucially,
since the non-Abelian anyons are engineered through defect lines rather than as
excitations, non-Abelian error correction is not required. Instead the error
correction problem is performed on the underlying Abelian model, allowing high
noise thresholds to be realized.
