  If a quantum dot is coupled to a topological superconductor via tunneling
contacts, each contact hosts a Majorana zero mode in the limit of zero
transmission. Close to a resonance and at a finite contact transparency, the
resonant level in the quantum dot couples the Majorana modes, but a ground
state degeneracy per fermion parity subspace remains if the number of Majorana
modes coupled to the dot is five or larger. Upon varying shape-defining gate
voltages while remaining close to resonance, a nontrivial evolution within the
degenerate ground-state manifold is achieved. We characterize the corresponding
non-Abelian holonomy for a quantum dot with chaotic classical dynamics using
random matrix theory and discuss measurable signatures of the non-Abelian
time-evolution.
