  As a signal recovery algorithm, compressed sensing is particularly useful
when the data has low-complexity and samples are rare, which matches perfectly
with the task of quantum phase estimation (QPE). In this work we present a new
Heisenberg-limited QPE algorithm for early quantum computers based on
compressed sensing. More specifically, given many copies of a proper initial
state and queries to some unitary operators, our algorithm is able to recover
the frequency with a total runtime
$\mathcal{O}(\epsilon^{-1}\text{poly}\log(\epsilon^{-1}))$, where $\epsilon$ is
the accuracy. Moreover, the maximal runtime satisfies $T_{\max}\epsilon \ll
\pi$, which is comparable to the state of art algorithms, and our algorithm is
also robust against certain amount of noise from sampling. We also consider the
more general quantum eigenvalue estimation problem (QEEP) and show numerically
that the off-grid compressed sensing can be a strong candidate for solving the
QEEP.
