Quantum algorithms for optimization problems are of general interest. Despite recent progress in classical lower bounds for nonconvex optimization under different settings and quantum lower bounds for convex optimization, quantum lower bounds for nonconvex optimization are still widely open. In this paper, we conduct a systematic study of quantum query lower bounds on finding $\epsilon$-approximate stationary points of nonconvex functions, and we consider the following two important settings: 1) having access to $p$-th order derivatives; or 2) having access to stochastic gradients. The classical query lower bounds is $\Omega\big(\epsilon^{-\frac{1+p}{p}}\big)$ regarding the first setting, and $\Omega(\epsilon^{-4})$ regarding the second setting (or $\Omega(\epsilon^{-3})$ if the stochastic gradient function is mean-squared smooth). In this paper, we extend all these classical lower bounds to the quantum setting. They match the classical algorithmic results respectively, demonstrating that there is no quantum speedup for finding $\epsilon$-stationary points of nonconvex functions with $p$-th order derivative inputs or stochastic gradient inputs, whether with or without the mean-squared smoothness assumption. Technically, our quantum lower bounds are obtained by showing that the sequential nature of classical hard instances in all these settings also applies to quantum queries, preventing any quantum speedup other than revealing information of the stationary points sequentially.

翻译：量子优化算法具有广泛的研究价值。尽管近年来在不同设置下的非凸优化经典下界以及凸优化量子下界取得了进展，非凸优化的量子下界问题仍普遍悬而未决。本文系统研究了寻找非凸函数$\epsilon$-近似驻点的量子查询下界，并考虑以下两种重要设置：1）可访问$p$阶导数；或2）可访问随机梯度。针对第一种设置，经典查询下界为$\Omega\big(\epsilon^{-\frac{1+p}{p}}\big)$；针对第二种设置，经典下界为$\Omega(\epsilon^{-4})$（若随机梯度函数满足均方光滑性，则为$\Omega(\epsilon^{-3})$）。本文将所有这些经典下界推广至量子场景。它们分别与经典算法结果相匹配，表明无论是否具备均方光滑性假设，对于通过$p$阶导数输入或随机梯度输入寻找非凸函数$\epsilon$-驻点的问题，均不存在量子加速。技术层面上，我们的量子下界通过证明上述所有设置中经典困难实例的序列特性同样适用于量子查询而获得，从而阻止了除顺序揭示驻点信息之外的任何量子加速。