Accurate Vibrational Frequency Calculations for Quantum Computing via an Analytic Second-Order Energy Derivative Framework

Quantum computing has emerged as a promising paradigm for tackling electronic structure problems, with most efforts to date focused on molecular energies and, more recently, first-order derivatives. However, the extension to second-order energy derivatives with respect to nuclear coordinates─essential for predicting vibrational spectra and identifying transition states─has remained relatively limited. Here, we present an analytic implementation for computing nuclear Hessians within the variational quantum eigensolver framework. Our approach produces harmonic vibrational frequencies and normal modes in excellent agreement with full configuration interaction (FCI) benchmarks, even for systems with challenging cases of orbital degeneracy or involving weak intermolecular interactions. We further assess the quantum measurement cost of these second-order derivative calculations and compare it with that of variational quantum eigensolver (VQE) energy calculations. Additionally, we demonstrate how point group symmetry can be incorporated to reduce measurement cost without loss of accuracy. This work extends quantum computing capabilities toward advanced quantum chemical simulations, such as the characterization of noncovalent interactions via low-frequency vibrational signatures and the identification of transition states.