von Neumann measurement and quantum phase estimation of block-encoded Hamiltonians

We review how to use von Neumann's measurement procedure to estimate a phase, using an efficient Hamiltonian simulation subroutine acts on a block-encoded Hamiltonian. We show that the resulting algorithm can be used to solve quantum phase estimation (QPE) or quantum energy estimation (QEE) {with competitive complexity scaling.} We then use recent results for block-encoding implementations to derive the Clifford + T complexity bound for QPE with respect to model-relevant parameters of the Hamiltonian and the desired precision. With this result, we demonstrate an efficient algorithm for QEE beginning from any linear combinations of Pauli strings. In this way, we argue that a well-understood and long-standing idea retains practical legitimacy for fault-tolerant era algorithms, once the costs of Hamiltonian simulation are accounted for.