We will introduce a novel hybrid quantum-classical quantum algorithm for solving many-body systems. An alternative to the well-known variational quantum eigensolvers, the new family of solvers, known as contracted quantum eigensolvers (CQE), minimize the residual of a contraction (or projection) of the Schrödinger equation onto the space of two (or more) electrons. The CQE is the quantum analog of classical methods for solving the anti-Hermitian contracted Schrödinger equation for the energies and two-electron reduced density matrices of ground and excited states without the many-electron wave function. The solver does not require deep circuits or difficult classical optimization and achieves a potentially exponential speed-up over its classical counterpart. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units. We apply the CQE with novel error-mitigation strategies on an IBM quantum computer to resolve the ground-state energies of the ortho-, meta-, and para- isomers of benzyne. Finally, the statically correlated active-space energy and two-electron reduced density matrix (2-RDM) from the CQE, we show, can be corrected to reflect the total electron correlation through a novel use of classical 2-RDM methods. Results will be presented for two different 2-RDM methods, the anti-Hermitian contracted Schrödinger equation (ACSE) and multi-component pair density functional (MC-PDFT) theories.
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