  We study the stability of the topological quantum computation proposals
involving Majorana fermions against thermal fluctuations. We use a minimal
realistic model of a spinless px+ipy superconductor and consider effect of
excited midgap states localized in the vortex core as well as of transitions
above the bulk superconducting gap on the quasiparticle braiding,
interferometry-based qubit read-out schemes, and quantum coherence of the
topological qubits. We find that thermal occupation of the midgap states does
not affect adiabatic braiding operations but leads to a reduction in the
visibility of the interferometry measurements. We also consider quantum
decoherence of topological qubits at finite temperatures and calculate their
decay rate which is associated with the change of the fermion parity and, as
such, is exponentially suppressed at temperatures well below the bulk
excitation gap. Our conclusion is that the Majorana-based topological quantum
computing schemes are indeed protected by the virtue of the quantum
non-locality of the stored information and the presence of the bulk
superconducting gap.
