Quantum algorithm for spectral projection by measuring an ancilla iteratively

We propose a quantum algorithm for projecting a quantum system to eigenstates of any Hermitian operator, provided one can access the associated control-unitary evolution for the ancilla and the system. The procedure is iterative by preparing a fresh ancilla state, applying the controlled unitary, and then measuring the ancilla. There are some freedoms in the ancilla state parameter and the control unitary. We give examples to illustrate that the alogrithm works. Simulations of the procedure also show that 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 also simulate this for a quantum spin chain.