  Quantum-enhanced auxiliary field quantum Monte Carlo (QC-AFQMC) uses output
from a quantum computer to increase the accuracy of its classical counterpart.
The algorithm requires the estimation of overlaps between walker states and a
trial wavefunction prepared on the quantum computer. We study the applicability
of this algorithm in terms of the number of measurements required from the
quantum computer and the classical costs of post-processing those measurements.
We compare the classical post-processing costs of state-of-the-art measurement
schemes using classical shadows to determine the overlaps and argue that the
overall post-processing cost stemming from overlap estimations scales like
$\mathcal{O}(N^9)$ per walker throughout the algorithm. With further numerical
simulations, we compare the variance behavior of the classical shadows when
randomizing over different ensembles, e.g., Cliffords and (particle-number
restricted) matchgates beyond their respective bounds, and uncover the
existence of covariances between overlap estimations of the AFQMC walkers at
different imaginary time steps. Moreover, we include analyses of how the error
in the overlap estimation propagates into the AFQMC energy and discuss its
scaling when increasing the system size.
