  Fermionic reduced density matrices summarize the key observables in fermionic
systems. In electronic systems, the two-particle reduced density matrix (2-RDM)
is sufficient to determine the energy and most physical observables of
interest. Here, we consider the possibility of using matrix completion to
reconstruct the two-particle reduced density matrix to chemical accuracy from
partial information. We consider the case of noiseless matrix completion, where
the partial information corresponds to a subset of the 2-RDM elements, as well
as noisy completion, where the partial information corresponds to both a subset
of elements, as well as statistical noise in their values. Through experiments
on a set of 24 molecular systems, we find that the 2-RDM can be efficiently
reconstructed from a reduced amount of information. In the case of noisy
completion, this results in multiple orders of magnitude reduction in the
number of measurements needed to determine the 2-RDM to chemical accuracy.
These techniques can be readily applied to both classical and quantum
algorithms for quantum simulations.
