Variational quantum computation with discrete variable representation for ro-vibrational calculations

Abstract: We demonstrate an approach to computing the vibrational energy levels of molecules that combines the discrete variable representation (DVR) of molecular Hamiltonians with variational quantum eigensolvers (VQE) and a greedy search of optimal quantum gate sequences. We show that the structure of the DVR Hamiltonians reduces the quantum measurement complexity scaling from exponential to polynomial, allowing for efficient VQE without second quantization. We then demonstrate that DVR Hamiltonians also lead to very efficient quantum ansatze for representing ro-vibrational states of molecules by states of a quantum computer. To obtain these compact representations, we demonstrate the quantum ansatz search by computing the vibrational energy levels of Cr$_2$ in seven electronic states as well as of van der Waals complexes Ar-HCl and Mg-NH. Our numerical results show that accuracy of 1~cm$^{-1}$ can be achieved by very shallow circuits with 2 to 9 entangling gates.