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

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