  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.
