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 Schrödinger equation (ACSE) theory and multicon Quantum-Classical Hybrid Algorithm for the Simulation of All-Electron Correlation Jan-Niklas Boyn, Aleksandr O. Lykhin, Scott E. Smart, Laura Gagliardi, ∗ and David A. Mazziotti∗ Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, IL 60637 E-mail: lgagliardi@uchicago.edu; damazz@uchicago.edu Abstract While the treatment of chemically relevant systems containing hundreds or even thou- sands of electrons remains beyond the reach of quantum devices, the development of quantum- classical hybrid algorithms to resolve electronic correlation presents a promising pathway to- ward 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 pro- posed, due to the dependence on the wave function of most classical methods for dynamic cor- relation, 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 calcula- tions 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 effi- ciently measured in the quantum simulation. Although the algorithm is completely general, we 1 arXiv:2106.11972v1 [quant-ph] 22 Jun 2021 methods to generate a system-wide correlated 2-RDM, spanning all of the electrons and orbitals in the calculation. Importantly, the correlated 2-RDM recovers the all-electron correlation en- ergy and properties of a molecule, thereby enabling larger basis sets and realistic comparisons with experimental results. While other hybrid algorithms use a “perturb-then-diagnalonize” strat- egy to add some dynamic correlation to the Hamiltonian before simulation, most algorithms have avoided the conventional “diagonalize-then-perturb” strategy because of the wave function bottle- neck.2,3,10,12–19 Although the algorithm is completely general for any 2-RDM-like methods, we test it here with two classical correlation methods: ( i) the anti-Hermitian contracted Schrödinger equation (ACSE) theory in which total correlation is computed from a functional of the 2-RDM that is seeded with the statically correlated (47) Mazziotti, D. A. Multireference many-electron correlation energies from two-electron re- duced density matrices computed by solving the anti-Hermitian contracted Schrödinger equa- tion. Phys. Rev. A 2007, 76, 052502. (48) Snyder, J. W.; Mazziotti, D. A. Conical Intersection of the Ground and First Excited States of Water: Energies and Reduced Density Matrices from the Anti-Hermitian Contracted Schrödinger Equation. J. Phys. Chem. A 2011, 115, 14120–14126. (49) Snyder, J. W.; Mazziotti, D. A. Photoexcited conversion of gauche-1,3-butadiene to bicy- clobutane via a conical intersection: Energies and reduced density matrices from the anti- Hermitian contracted Schrödinger equation. J. Chem. Phys. 2011, 135, 024107. (50) Smart, S. E.; Scrape, P. G.; Butler, L. J.; Mazziotti, D. A. Using reduced density matrix techniques to capture static and dynamic correlation in the energy landscape for the decom- position of the CH2CH2ONO radical and support a non-IRC pathway. J. Chem. Phys. 2018, 149, 024302. (51) Gidofalvi, G.; Mazziotti, D. A. Direct calculation of excited-state electronic energies and two- electron reduced density matrices from the anti-Hermitian contracted Schrödinger equation. Phys. Rev. A 2009, 80, 022507. (52) Sturm, E. J.; Mazziotti, D. A. Highly accurate excited-state energies from direct computa- tion of the 2-electron reduced density matrix by the anti-Hermitian contracted Schrödinger equation. Mol. Phys. 2016, 114, 335–343. (53) Bonet-Monroig, X.; Babbush, R.; O’Brien, T. E. Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States. Phys. Rev. X 2020, 10, 031064. (54) Smart, S. E.; Mazziotti, D. A. Lowering tomography costs in quantum simulation with a symmetry projected operator basis. Phys. Rev. A 2021, 103, 012420. (55) IBM Quantum. 2020. 24