  Natural frequencies and normal modes are basic properties of a structure
which play important roles in analyses of its vibrational characteristics. As
their computation reduces to solving eigenvalue problems, it is a natural arena
for application of quantum phase estimation algorithms, in particular for large
systems. In this note, we take up some simple examples of (classical) coupled
oscillators and show how the algorithm works by using qubitization methods
based on a sparse structure of the matrix. We explicitly construct
block-encoding oracles along the way, propose a way to prepare initial states,
and briefly touch on a more generic oracle construction for systems with
repetitive structure. As a demonstration, we also give rough estimates of the
necessary number of physical qubits and actual runtime it takes when carried
out on a fault-tolerant quantum computer.
