<p>Non-Abelian braiding operations, a fundamental mechanism for implementing topological quantum gate operations, have recently been simulated using classical acoustic and photonic waves. However, such operations relied on adiabatic non-Abelian holonomies, which render the practical implementation intrinsically slow. Here, we theoretically propose and experimentally demonstrate a simulation of non-adiabatic non-Abelian braiding operations using quantum matter waves in a Bose-Einstein condensate. Within our framework, the effective braid strands are encoded by the momentum states, while the simulated braiding transformations arise from non-adiabatic non-Abelian holonomies. Experimentally, we verify the non-Abelian algebra by showing that different sequences of holonomic operations, applied to the same initial momentum state, yield distinct outcomes. Furthermore, we exhibit that these operations can be leveraged to prepare, transfer and distribute momentum quantum superposition states. Our work opens a fast non-adiabatic paradigm for non-Abelian geometric manipulation of both classical and quantum waves, and brings non-Abelian braiding operation into a regime applicable to quantum state manipulation.</p>