Quantum algorithm for spectral projection by measuring an ancilla iteratively

We propose a quantum algorithm for projecting to eigenstates of any hermitian operator, provided one can access the associated control-unitary evolution and measurement of the ancilla of the control. The procedure is iterative and the distribution of the projected eigenstates obeys the Born rule. This algorithm can be used as a subroutine in the quantum annealing procedure by measurement to drive the system to the ground state, and we demonstrate its feasibility by simulating the procedure