  Recently, feedback-based quantum algorithms have been introduced to calculate
the ground states of Hamiltonians, inspired by quantum Lyapunov control theory.
This paper aims to generalize these algorithms to the problem of calculating an
eigenstate of a given Hamiltonian, assuming that the lower energy eigenstates
are known. To this aim, we propose a new design methodology that combines the
layer-wise construction of the quantum circuit in feedback-based quantum
algorithms with a new feedback law based on a new Lyapunov function to assign
the quantum circuit parameters. We present two approaches for evaluating the
circuit parameters: one based on the expectation and overlap estimation of the
terms in the feedback law and another based on the gradient of the Lyapunov
function. We demonstrate the algorithm through an illustrative example and
through an application in quantum chemistry. To assess its performance, we
conduct numerical simulations and execution on IBM's superconducting quantum
computer.
