  The Kitaev chain model exhibits topological order that manifests as
topological degeneracy, Majorana edge modes and $Z_{2}$ topological invariance
of the abulk spectrum. This model can be obtained from a transverse field Ising
model(TFIM) using the Jordan-Wigner transformation. TFIM has neither
topological degeneracy nor any edge modes. Topological degeneracy associated
with topological order is central to topological quantum computation. In this
paper we will explore topological protection of the ground state manifold in
the case of Majorana fermion models which exhibit $Z_{2}$ topological order. We
will show that there are at least two different ways to understand this
topological protection of Majorana fermion qubits: one way is based on
fermionic mode operators and the other is based on anti-commuting symmetry
operators. We will also show how these two different ways are related to each
other. We provide a very general approach of understanding the topological
protection of Majorana fermion qubits in the case of lattice Hamiltonians.
