  While quantum computing algorithms have been successfully applied to
electronic structure of molecules, applications of quantum computing to
molecular dynamics remain scarce. The variability of intra-molecular
interaction potentials gives rise to vibrational states with a wide range of
properties. It is therefore challenging to obtain a general representation of
molecular ro-vibrational states by states of a quantum computer with a limited
number of qubits and quantum gates. We demonstrate a general approach to
computing the ro-vibrational energy levels of molecules by combining the
discrete variable representation of molecular Hamiltonians with variational
quantum eigensolvers and a greedy search of gate permutations, yielding
accurate representations of both ground and excited vibrational states by
compact quantum circuits with a small number of gates. To illustrate the
generality and adaptability of this approach, we compute the vibrational energy
levels of Cr$_2$ in seven electronic states as well as the vibrational energy
levels of van der Waals complexes Ar-HCl and Mg-NH, illustrating that accuracy
of 1 cm$^{-1}$ can be achieved with between 2 and 9 entangling gates.
