  It is difficult to calculate the energy levels and eigenstates of a large
physical system on a classical computer because of the exponentially growing
size of the Hilbert space. In this work, we experimentally demonstrate a
quantum algorithm which could solve this problem via simulated resonant
transitions. Using a four-qubit quantum simulator in which two qubits are used
as ancillas for control and measurement, we obtain the energy spectrum of a
2-qubit low-energy effective Hamiltonian of the water molecule. The simulated
transitions allow the state of the quantum simulator to transform and access
large regions of the Hilbert space, including states that have no overlap with
the initial state. Furthermore, we make use of this algorithm to efficiently
prepare specific eigenstates on the simulator according to the measured
eigenenergies.
