  We unravel the Rodeo Algorithm to determine the eigenstates and eigenvalues
spectrum for a general Hamiltonian considering arbitrary initial states. By
presenting a novel methodology, we detail the original method and show how to
define all properties without having prior knowledge regarding the eigenstates.
To this end, we exploit Pennylane and Qiskit platforms resources to analyze
scenarios where the Hamiltonians are described by the Zeeman model for one and
two spins. We also introduce strategies and techniques to improve the
algorithm's performance by adjusting its intrinsic parameters and reducing the
fluctuations inherent to data distribution. First, we explore the dynamics of a
single qubit on Xanadu simulators to set the parameters that optimize the
method performance and select the best strategies to execute the algorithm. On
the sequence, we extend the methodology for bipartite systems to discuss how
the algorithm works when degeneracy and entanglement are taken into account.
Finally, we compare the predictions with the results obtained on a real
superconducting device provided by the IBM Q Experience program, establishing
the conditions to increase the protocol efficiency for multi-qubit systems.
