  Accurate simulations of vibrational molecular spectra are expensive on
conventional computers. Compared to the electronic structure problem, the
vibrational structure problem with quantum computers is less investigated. In
this work we accurately estimate quantum resources, such as number of qubits
and quantum gates, required for vibrational structure calculations on a
programmable quantum computer. Our approach is based on quantum phase
estimation and focuses on fault-tolerant quantum devices. In addition to
asymptotic estimates for generic chemical compounds, we present a more detailed
analysis of the quantum resources needed for the simulation of the Hamiltonian
arising in the vibrational structure calculation of acetylene-like polyynes of
interest. Leveraging nested commutators, we provide an in-depth quantitative
analysis of trotter errors compared to the prior investigations. Ultimately,
this work serves as a guide for analyzing the potential quantum advantage
within vibrational structure simulations.
