In this work we propose a new hybrid quantum algorithm combining the Variational Quan- tum Eigensolver (VQE) algorithm with a classical embedding method. The embedding consists in employing Multiconfiguration Pair Density Functional Theory (MC-PDFT) theory as a post-processing method for calculating the final energy of the system, after obtaining the ground state wave function from the quantum computer. In this way, we are able to treat to exponentially scaling part of the problem with VQE, with the promise of a speed up in the computation. The hybrid algorithm is motivated by the need of prop- erly treating electron correlation in strongly correlated systems. In out case, VQE mainly takes care of the static part of the correlation while the MC-PDFT embedding allows to recover most of the dynamic correlation. The MC-PDFT theory combines wave function calculations with Density Functional Theory (DFT). We also propose a self-consistently optimized version of VQE, where molecular coefficient are re-optimized after every itera- tion until convergence. In this way, we are able to achieve better accuracy in the resulting energy.

By comparing the performance of our algorithm with classical methods having similar computational costs, we observe an improvement in the accuracy of dissociation energies and ground state energies of different molecules. We test our embedding method with H2, H2O and N2. Our aim is not only to enhance the precision of VQE in predicting molecular energies, but also to reduce the depth and the width of the quantum circuit. The classical embedding allows us to alleviate quantum resources by treating part of the dynamical correlation. Since MC-PDFT consists in an active space (AS) calculation, the number of orbitals required is lower and so is the number of qubits. We also achieve a significant reduction in the gate count by combining the MC-PDFT embedding and the ADAPT-VQE algorithm. Results show a reduction in the number of CNOTs by more than half compared to the initial count in some cases. Overall, this algorithm is able to achieve, in most cases, an accuracy comparable to more computationally expensive clas- sical methods and to reduce quantum resources while maintaining a reasonable precision in the results. We believe that this method can pave the way toward the simulation of larger molecular systems with current quantum devices.