Quantum Lower Bounds for Finding Stationary Points of Nonconvex Functions

TL;DR

This paper establishes quantum query lower bounds for finding approximate stationary points in nonconvex optimization, showing no quantum speedup over classical methods in these settings.

Contribution

It extends classical lower bounds to the quantum setting for nonconvex optimization, demonstrating the limits of quantum speedup in this area.

Findings

Quantum lower bounds match classical bounds, indicating no quantum speedup.

Sequential nature of classical hard instances applies to quantum queries.

Quantum algorithms cannot outperform classical algorithms for these nonconvex problems.

Abstract

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.