  While the treatment of chemically relevant systems containing hundreds or
even thousands of electrons remains beyond the reach of quantum devices, the
development of quantum-classical hybrid algorithms to resolve electronic
correlation presents a promising pathway toward a quantum advantage in the
computation of molecular electronic structure. Such hybrid algorithms treat the
exponentially scaling part of the calculation -- the static (multireference)
correlation -- on the quantum computer and the non-exponentially scaling part
-- the dynamic correlation -- on the classical computer. While a variety of
such algorithms have been proposed, due to the dependence on the wave function
of most classical methods for dynamic correlation, the development of
easy-to-use classical post-processing implementations has been limited. Here we
present a novel hybrid-classical algorithm that computes a molecule's
all-electron energy and properties on the classical computer from a critically
important simulation of the static correlation on the quantum computer.
Significantly, for the all-electron calculations we circumvent the wave
function by using density-matrix methods that only require input of the
statically correlated two-electron reduced density matrix (2-RDM), which can be
efficiently measured in the quantum simulation. Although the algorithm is
completely general, we test it with two classical 2-RDM methods, the
anti-Hermitian contracted Schr\"odinger equation (ACSE) theory and
multiconfiguration pair-density functional theory (MC-PDFT), using the recently
developed quantum ACSE method for the simulation of the statically correlated
2-RDM. We obtain experimental accuracy for the relative energies of all three
benzyne isomers and thereby, demonstrate the ability of the quantum-classical
hybrid algorithms to achieve chemically relevant results and accuracy on
currently available quantum computers.
