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

## TL;DR

This paper explores the topological protection mechanisms of Majorana fermion qubits in lattice Hamiltonians, highlighting two approaches based on fermionic modes and symmetry operators, and their interrelation.

## Contribution

It introduces a general framework for understanding topological protection of Majorana qubits, connecting fermionic mode and symmetry operator perspectives.

## Key findings

- Topological degeneracy and edge modes in the Kitaev chain model.
- Two distinct methods to understand topological protection.
- Relation between fermionic modes and symmetry operators in protection mechanisms.

## Abstract

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.