Robustness of Quantum Algorithms for Nonconvex Optimization

TL;DR

This paper investigates the robustness of quantum algorithms for nonconvex optimization, demonstrating their ability to find approximate stationary points under noisy conditions with polynomial speedups over classical methods.

Contribution

It provides the first systematic analysis of quantum algorithms' robustness to noise in nonconvex optimization, introducing new algorithms and bounds for noisy quantum oracle settings.

Findings

Quantum algorithms can find approximate stationary points with noise tolerance up to certain bounds.

Quantum algorithms achieve polynomial speedup over classical counterparts in noisy nonconvex optimization.

The paper characterizes domains where quantum algorithms are efficient or infeasible based on query complexity.

Abstract

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.