In the current era of noisy quantum devices, there is a need for quantum algorithms that are efficient and robust against noise. Towards this end, we introduce the projected cooling algorithm for quantum computation. The projected cooling algorithm is able to construct the localized ground state of any Hamiltonian with a translationally-invariant kinetic energy. The method can be viewed as the quantum analog of evaporative cooling. We start with an initial state with support over a compact region of a large volume. We then drive the excited quantum states to disperse and measure the remaining portion of the wave function left behind. The method can be used in concert with other techniques such as variational methods and adiabatic evolution to achieve better performance than existing approaches for the same number of quantum gates per qubit. For the nontrivial examples we consider here, the improvement is substantial. The only additional resource required is performing the operations in a volume significantly larger than the size of the localized state.