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The Kitaev chain model exhibits topological order that manifests astopological degeneracy, Majorana edge modes and $Z_{2}$ topological invarianceof the abulk spectrum. This model can be obtained from a transverse field Isingmodel(TFIM) using the Jordan-Wigner transformation. TFIM has neithertopological degeneracy nor any edge modes. Topological degeneracy associatedwith topological order is central to topological quantum computation. In thispaper we will explore topological protection of the ground state manifold inthe case of Majorana fermion models which exhibit $Z_{2}$ topological order. Wewill show that there are at least two different ways to understand thistopological protection of Majorana fermion qubits: one way is based onfermionic mode operators and the other is based on anti-commuting symmetryoperators. We will also show how these two different ways are related to eachother. We provide a very general approach of understanding the topologicalprotection of Majorana fermion qubits in the case of lattice Hamiltonians.

$Z_{2}$ Topological Order and Topological Protection of Majorana Fermion Qubits