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

Classical and quantum cost of measurement strategies for quantum-enhanced auxiliary field Quantum Monte Carlo