Robustness of Quantum Algorithms for Nonconvex Optimization

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 $ε$-approximate second-order stationary point ($ε$-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 $O(ε^{10}/d^5)$, accelerated perturbed gradient descent with quantum gradient estimation takes $O(\log d/ε^{1.75})$ quantum queries to find an $ε$-SOSP. We then prove that perturbed gradient descent is robust to the noise of $O(ε^6/d^4)$ and $O(ε/d^{0.5+ζ})$ for $ζ>0$ 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 $O(ε^{1.5}/d)$ and $O(ε/\sqrt{d})$ noise on the zeroth- and first-order oracles, respectively. The quantum algorithm takes $O(d^{2.5}/ε^{3.5})$ and $O(d^2/ε^3)$ queries to the two oracles, giving a polynomial speedup over the classical counterparts. Moreover, we characterize the domains where quantum algorithms can find an $ε$-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 $Ω(ε^{-12/7})$ lower bound in $ε$ for any randomized classical and quantum algorithm to find an $ε$-SOSP using either noisy zeroth- or first-order oracles.