非凸优化量子算法的鲁棒性
Robustness of Quantum Algorithms for Nonconvex Optimization
摘要：
最近的研究结果表明，量子计算机具有加速非凸优化问题的潜力。然而，实现量子优化算法的一个关键因素是它们对实验和统计噪声的鲁棒性。在本文中，我们系统地研究量子算法寻找$d$ -维非凸函数的$ε$ -近似二阶驻点（$ε$ -SOSP），非凸优化的基本问题，噪声零阶或一阶预言机作为输入。我们首先证明，直到$O(ε^{10}/d^5)$的噪声，加速扰动梯度下降与量子梯度估计需要$O(\log d/ε^{1.75})$量子查询找到一个$ε$-SOSP。然后，我们证明了扰动梯度下降是鲁棒的噪声的$O(ε^6/d^4)$和$O(ε/d^{0.5+ζ})$的$ζ>0$上的零阶和一阶的预言机，分别提供了一个量子算法与多对数查询复杂度。然后，我们提出了一个随机梯度下降算法，使用量子均值估计的高斯平滑的噪声预言机，这是强大的$O(ε^{1.5}/d)$和$O(ε/\sqrt{d})$噪声的零阶和一阶预言机，分别。量子算法将$O(d^{2.5}/ε^{3.5})$和$O(d^2/ε^3)$查询传递给两个预言机，相对于经典算法给出了多项式加速。此外，我们的特点域量子算法可以找到一个$ε$ -SOSP与多对数，多项式，或指数的查询$d$，或问题是信息理论上无法解决的，甚至由无限数量的查询。此外，我们证明了一个$Ω(ε^{-12/7})$下界在$ε$的任何随机化的经典和量子算法找到一个$ε$-SOSP使用噪声零阶或一阶预言。
英文摘要：
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.
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