  We consider topological quantum memories for a general class of abelian anyon
models defined on spin lattices. These are non-universal for quantum
computation when restricting to topological operations alone, such as braiding
and fusion. The effects of additional non-topological operations, such as spin
measurements, are studied. These are shown to allow universal quantum
computation, while still utilizing topological protection. Our work gives an
insight into the relation between abelian models and their non-abelian
counterparts.
