Resource-frugal Hamiltonian eigenstate preparation via repeated quantum phase estimation measurements

The preparation of Hamiltonian eigenstates is essential for many applications in quantum computing; the efficiency with which this can be done is of key interest. A canonical approach exploits the quantum phase estimation (QPE) algorithm. We adopt ideas from variants of this method to implement a resource-frugal iterative scheme, and provide analytic bounds on the complexity (simulation time cost) for various cases of available information and tools. We propose and characterise an extension involving a modification of the target Hamiltonian to increase overall efficiency. The presented methods and bounds are then demonstrated by preparing the ground state of the Hamiltonians of LiH and H$_2$ in second quantisation; we report the performance of both ideal and noisy implementations using simulated quantum computers. Convergence is generally achieved much faster than the bounds suggest, while the qualitative features are validated.