  The evaluation of expectation values $Tr\left[\rho O\right]$ for some pure
state $\rho$ and Hermitian operator $O$ is of central importance in a variety
of quantum algorithms. Near optimal techniques developed in the past require a
number of measurements $N$ approaching the Heisenberg limit
$N=\mathcal{O}\left(1/\epsilon\right)$ as a function of target accuracy
$\epsilon$. The use of Quantum Phase Estimation requires however long circuit
depths $C=\mathcal{O}\left(1/\epsilon\right)$ making their implementation
difficult on near term noisy devices. The more direct strategy of Operator
Averaging is usually preferred as it can be performed using
$N=\mathcal{O}\left(1/\epsilon^2\right)$ measurements and no additional gates
besides those needed for the state preparation.
  In this work we use a simple but realistic model to describe the bound state
of a neutron and a proton (the deuteron) and show that the latter strategy can
require an overly large number of measurement in order to achieve a reasonably
small relative target accuracy $\epsilon_r$. We propose to overcome this
problem using a single step of QPE and classical post-processing. This approach
leads to a circuit depth $C=\mathcal{O}\left(\epsilon^\mu\right)$ (with
$\mu\geq0$) and to a number of measurements
$N=\mathcal{O}\left(1/\epsilon^{2+\nu}\right)$ for $0<\nu\leq1$. We provide
detailed descriptions of two implementations of our strategy for $\nu=1$ and
$\nu\approx0.5$ and derive appropriate conditions that a particular problem
instance has to satisfy in order for our method to provide an advantage.
