Obtaining the energy levels and eigenstates of a large physical system is a difficult task on a classical computer because of the exponentially growing size of the Hilbert space. We proposed a quantum algorithm which solves this problem via a mechanism of quantum simulation of resonant transitions (QSRT). The simulated transitions allow the state of a 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 according to the measured eigenenergies.

We will also discuss QSRT in multi-steps, which gives the flexibility in designing an evolution path from an initial Hamiltonian to the system Hamiltonian. This approach provides an efficient way for solving problems with a special structure.