  Among the list of major threats to quantum computation, quantum decoherence
poses one of the largest because it generates losses to the environment within
a computational system which cannot be recovered via error correction methods.
These methods require the assumption that the environmental interaction forces
the qubit state into some linear combination of qubit eigenstates. In reality,
the environment causes the qubit to enter into a mixed state where the original
is no longer recoverable. A promising solution to this problem bases the
computational states on the low lying energy excitations within topological
materials. The existence of these states is protected by a global parameter
within the Hamiltonian which prevents the computational states from coupling
locally and decohering. In this paper, the qubit is based on nonlocal,
topological Majorana fermions (MF), and the gate operations are generated by
swapping or braiding the positions of said MF. The algorithmic calculation for
such gate operations is well known, but, the opposite gates-to-braid
calculation is currently underdeveloped. Additionally, because one may choose
from a number of different possible qubit definitions, the resultant gate
operations from calculation to calculation appear different. Here, the
calculations for the two- and four-MF cases are recapitulated for the sake of
logical flow. This set of gates serves as the foundation for the understanding
and construction of the six-MF cases. Using these, a full characterization of
the system is made by completely generalizing the list of gates and
transformations between possible qubit definitions. A complete description of
this system is desirable and will hopefully serve future iterations of
topological qubits.
