Majorana zero modes are emergent zero-energy quasiparticle excitations in certain superconducting

systems that can be viewed as fractionalized or “half” electrons. These

quasiparticles obey non-Abelian braiding statistics which is one manifestation of such

half-electron character. Due to such non-Abelian braiding property, Majorana zero

mode pairs hold promise as potential qubits for topological quantum computation.

It is somewhat surprising that, at least theoretically, ordinary one-dimensional semiconductor

systems can be induced to host such esoteric Majorana modes as edge states

if some precise experimental conditions are satisfied. Because of the relative simplicity

of material and experimental requirements to host Majorana modes, there has been a

flurry of experimental effort to realize them in semiconductor nanowire systems. While

experimental efforts have produced preliminary evidence for the presence of Majorana

zero modes in these systems, a thorough confirmation is lacking. The experimental signature

in question is the presence of a zero-bias conductance-peak that, while necessary,

is not a sufficient criterion to establish the presence of underlying Majorana modes. Given the importance of Majorana braiding for topological quantum computation and skepticism

over the presence of Majorana modes in these experimental systems, it would seem

natural to attempt braiding these putative Majorana modes in the near future. In that

case, an observation of non-Abelian statistics would provide the necessary and sufficient

condition in favor of Majorana presence in the studied experimental systems.

This thesis has three distinct parts. First, we assume perfect Majorana modes as

given that can be successfully braided. In this case, we calculate the diabatic error due

to the finite speed of braiding when the system is coupled to a Bosonic bath. Next, we

grant that the mechanism for zero-bias conductance-peak is indeed topological, albeit

the underlying Majorana modes may be imperfect (the modes are not precisely at zero

energy). We study the interplay of dissipation and finite energy splitting of the Majorana

modes and study its consequence regarding the probability of successful braiding.

Lastly, we propose studying the correlation between independent left and right conductance

measurements as a means to distinguish between a topological versus a non-topological

the mechanism underlying the observed zero-bias conductance-peak.