  We study the problem of estimating frequency response functions of systems of
coupled, classical harmonic oscillators using a quantum computer. The
functional form of these response functions can be mapped to a corresponding
eigenproblem of a Hermitian matrix $H$, thus suggesting the use of quantum
phase estimation. Our proposed quantum algorithm operates in the standard
$s$-sparse, oracle-based query access model. For a network of $N$ oscillators
with maximum norm $\lVert H \rVert_{\mathrm{max}}$, and when the eigenvalue
tolerance $\varepsilon$ is much smaller than the minimum eigenvalue gap, we use
$\mathcal{O}(\log(N s \lVert H \rVert_{\mathrm{max}}/\varepsilon)$ algorithmic
qubits and obtain a rigorous worst-case query complexity upper bound
$\mathcal{O}(s \lVert H \rVert_{\mathrm{max}}/(\delta^2 \varepsilon) )$ up to
logarithmic factors, where $\delta$ denotes the desired precision on the
coefficients appearing in the response functions. Crucially, our proposal does
not suffer from the infamous state preparation bottleneck and can as such
potentially achieve large quantum speedups compared to relevant classical
methods. As a proof-of-principle of exponential quantum speedup, we show that a
simple adaptation of our algorithm solves the random glued-trees problem in
polynomial time. We discuss practical limitations as well as potential
improvements for quantifying finite size, end-to-end complexities for
application to relevant instances.
