Advantage of discrete variable representation in variational quantum eigensolvers for vibrational energy calculations
While quantum computing algorithms have been widely applied for electronic structure calculations, applications to molecular dynamics remain scarce. Complex and varied landscapes of molecular potential energy surfaces give rise to vibrational states with a wide range of properties, making it difficult to construct a general representation of ro-vibrational states by a quantum computer with a limited number of qubits and gates. Another challenge is the exponential growth of the computation complexity - for example, the number of terms required to expand a general Hamiltonian in Pauli strings increases exponentially with the number of qubits. Here, we show that discrete variable representation (DVR) can be leveraged to represent molecular Hamiltonians by the polynomial (in the number of qubits) number of quantum circuits. We then demonstrate that DVR Hamiltonians lead to very efficient quantum ansatze for vibrational states. For this purpose, we develop a compositional quantum ansatz search that adapts gate sequences in variational quantum eigensolvers (VQE) to a specific molecular state. We apply VQE to compute 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 quantum circuits with 2 to 9 entangling gates.