Iterative phase estimation has long been used in quantum computing to estimate Hamiltonian eigenvalues. This is done by applying many repetitions of the same fundamental simulation circuit to an initial state, and using statistical inference to glean estimates of the eigenvalues from the resulting data. Here, we show a generalization of this framework where each of the steps in the simulation uses a different Hamiltonian. This allows the precision of the Hamiltonian to be changed as the phase estimation precision increases. Additionally, through the use of importance sampling, we can exploit knowledge about the ground state to decide how frequently each Hamiltonian term should appear in the evolution, and minimize the variance of our estimate. We rigorously show, if the Hamiltonian is gapped and the sample variance in the ground state expectation values of the Hamiltonian terms sufficiently small, that this process has a negligible impact on the resultant estimate and the success probability for phase estimation. We demonstrate this process numerically for two chemical Hamiltonians, and observe substantial reductions in the number of terms in the Hamiltonian; in one case, we even observe a reduction in the number of qubits needed for the simulation. Our results are agnostic to the particular simulation algorithm, and we expect these methods to be applicable to a range of approaches.
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