  Density matrix downfolding (DMD) is a technique for regressing low-energy
effective Hamiltonians from quantum many-body Hamiltonians. One limiting factor
in the accuracy of classical implementations of DMD is the presence of
difficult-to-quantify systematic errors attendant to sampling the observables
of quantum many-body systems on an approximate low-energy subspace. We propose
a hybrid quantum-classical protocol for circumventing this limitation, relying
on the prospective ability of quantum computers to efficiently prepare and
sample from states in well-defined low-energy subspaces with systematically
improvable accuracy. We introduce three requirements for when this is possible,
including a notion of compressibility that quantifies features of Hamiltonians
and low-energy subspaces thereof for which quantum DMD might be efficient.
Assuming that these requirements are met, we analyze design choices for our
protocol and provide resource estimates for implementing quantum-enhanced DMD
on both the doped 2-D Fermi-Hubbard model and an ab initio model of a cuprate
superconductor.
