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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.