Abstract: In recent decades, quantum computers have been postulated as an alternative with greater computational power than classical computers for solving certain problems. However, although some algorithms demonstrate exponential advantages in data processing, efficient data loading is still a reliable challenge. The absence of a general protocol to address this problem means that the coding method has to be adapted to the needs of the algorithm in question. In this work, we focus on one of the most important encodings, the dynamic or Hamiltonian embedding, specifically on a protocol that allows for the exponentiation of a density matrix ρ by employing multiple copies of it. Studying this protocol from an original quantum-channel perspective, we have been able to find a more general Hamiltonian than the swap operation proposed in previous literature. Remarkably, we have proven that the performance of the swap cannot be improved if the protocol is independent of the initial state, but it does admit improvement when a dependence on the density matrices is allowed. Last but not least, as density matrices are a very restricted family of square matrices, we have extended the protocol to general square matrices.