Topological quantum computation away from the ground state using Majorana fermions

We relax one of the requirements for topological quantum computation with Majorana fermions. Topological quantum computation was discussed so far as manipulation of the wave function within degenerate many-body ground state. The simplest particles providing degenerate ground state, Majorana fermions, often coexist with extremely low-energy excitations so keeping the system in the ground state may be hard. We show that the topological protection extends to the excited states, as long as the Majorana fermions do not interact neither directly nor via the excited states. This protection relies on the fermion parity conservation and so it is generic to any implementation of Majorana fermions.