  Inspired by non-abelian vortex anyons in spinor Bose-Einstein condensates, we
consider the quantum double $\mathcal{D}(Q_8)$ anyon model as a platform to
carry out a particular instance of Shor's factorization algorithm. We provide
the excitation spectrum, the fusion rules, and the braid group representation
for this model, and design a circuit architecture that facilitates the
computation. All necessary quantum gates, less one, can be compiled exactly for
this hybrid topological quantum computer, and to achieve universality the last
operation can be implemented in a non-topological fashion. To analyse the
effect of decoherence on the computation, a noise model based on stochastic
unitary rotations is considered. The computational potential of this quantum
double anyon model is similar to that of the Majorana fermion based Ising anyon
model, offering a complementary future platform for topological quantum
computation.
