  We propose and analyze a physical system capable of performing topological
quantum computation with Majorana zero modes (MZM) in a one-dimensional
topological superconductor (1DTS). One of the leading methods to realize
quantum gates in 1DTS is to use T-junctions, which allows one to maneuver MZMs
such as to achieve braiding. In this paper, we propose a scheme that is in a
purely one-dimensional geometry and does not require T-junctions, instead
replacing it with an auxiliary qubit. We show that this allows one to perform
one and two logical qubit $ Z $ rotations. We first design a topologically
protected logical $Z$-gate based entirely on local interactions within the
1DTS. Using an auxiliary qubit coupled to the topological superconductors, we
extend the $Z$-gate to single and multiqubit arbitrary rotations with partial
topological protection. Finally, to perform universal quantum computing, we
introduce a scheme for performing arbitrary unitary rotations, albeit without
topological protection. We develop a formalism based on unitary braids which
creates transitions between different topological phases of the 1DTS system.
The unitary formalism can be simply converted to an equivalent adiabatic
scheme, which we numerically simulate and show that high fidelity operations
should be possible with reasonable parameters.
