  Solving the electronic structure problem of molecules and solids to high
accuracy is a major challenge in quantum chemistry and condensed matter
physics. The rapid emergence and development of quantum computers offer a
promising route to systematically tackle this problem. Recent work by Huggins
et al.[1] proposed a hybrid quantum-classical quantum Monte Carlo (QC-QMC)
algorithm using Clifford shadows to determine the ground state of a Fermionic
Hamiltonian. This approach displayed inherent noise resilience and the
potential for improved accuracy compared to its purely classical counterpart.
Nevertheless, the use of Clifford shadows introduces an exponentially scaling
post-processing cost. In this work, we investigate an improved QC-QMC scheme
utilizing the recently developed Matchgate shadows technique [2], which removes
the aforementioned exponential bottleneck. We observe from experiments on
quantum hardware that the use of Matchgate shadows in QC-QMC is inherently
noise robust. We show that this noise resilience has a more subtle origin than
in the case of Clifford shadows. Nevertheless, we find that classical
post-processing, while asymptotically efficient, requires hours of runtime on
thousands of classical CPUs for even the smallest chemical systems, presenting
a major challenge to the scalability of the algorithm.
