We propose a scheme for performing Majorana zero mode (MZM) braiding utilizing cotunneling processes in a three-MZM system incorporating reference arms. This approach relies on the interference between cotunneling paths through the MZMs and reference arms, establishing an effective, tunable coupling between the MZMs. The strength and sign of this coupling can be manipulated via the reference arms and applied magnetic flux. Notably, the introduction of a half-quantum flux reverses the coupling sign, enabling an echolike protocol to eliminate dynamic phases during braiding. Our setup, requiring only three MZMs, represents a minimal platform for demonstrating non-Abelian braiding statistics. We demonstrate that this system facilitates the implementation of Clifford gates via braiding and, significantly, permits the realization of non-Clifford gates, such as the $T$ gate, by geometric phase, thereby offering a potential pathway toward universal topological quantum computation.

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• Anyons
• Geometric & topological phases
• Majorana bound states
• Topological quantum computing
• Topological superconductors
• Topological materials
• Lindblad equation