  Recent results suggest that quantum computers possess the potential to speed
up nonconvex optimization problems. However, a crucial factor for the
implementation of quantum optimization algorithms is their robustness against
experimental and statistical noises. In this paper, we systematically study
quantum algorithms for finding an $\epsilon$-approximate second-order
stationary point ($\epsilon$-SOSP) of a $d$-dimensional nonconvex function, a
fundamental problem in nonconvex optimization, with noisy zeroth- or
first-order oracles as inputs. We first prove that, up to noise of
$O(\epsilon^{10}/d^5)$, accelerated perturbed gradient descent with quantum
gradient estimation takes $O(\log d/\epsilon^{1.75})$ quantum queries to find
an $\epsilon$-SOSP. We then prove that perturbed gradient descent is robust to
the noise of $O(\epsilon^6/d^4)$ and $O(\epsilon/d^{0.5+\zeta})$ for $\zeta>0$
on the zeroth- and first-order oracles, respectively, which provides a quantum
algorithm with poly-logarithmic query complexity. We then propose a stochastic
gradient descent algorithm using quantum mean estimation on the Gaussian
smoothing of noisy oracles, which is robust to $O(\epsilon^{1.5}/d)$ and
$O(\epsilon/\sqrt{d})$ noise on the zeroth- and first-order oracles,
respectively. The quantum algorithm takes $O(d^{2.5}/\epsilon^{3.5})$ and
$O(d^2/\epsilon^3)$ queries to the two oracles, giving a polynomial speedup
over the classical counterparts. Moreover, we characterize the domains where
quantum algorithms can find an $\epsilon$-SOSP with poly-logarithmic,
polynomial, or exponential number of queries in $d$, or the problem is
information-theoretically unsolvable even by an infinite number of queries. In
addition, we prove an $\Omega(\epsilon^{-12/7})$ lower bound in $\epsilon$ for
any randomized classical and quantum algorithm to find an $\epsilon$-SOSP using
either noisy zeroth- or first-order oracles.
