# Quantum algorithms for escaping from saddle points

Chenyi Zhang1, Jiaqi Leng2, and Tongyang Li3,4,5

1Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
2Department of Mathematics and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA
3Center on Frontiers of Computing Studies, Peking University, Beijing, China
4Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
5Department of Computer Science and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA

### Abstract

We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function $f\colon\mathbb{R}^{n}\to\mathbb{R}$, our quantum algorithm outputs an $\epsilon$-approximate second-order stationary point using $\tilde{O}(\log^{2} (n)/\epsilon^{1.75})$ queries to the quantum evaluation oracle (i.e., the zeroth-order oracle). Compared to the classical state-of-the-art algorithm by Jin et al. with $\tilde{O}(\log^{6} (n)/\epsilon^{1.75})$ queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms of $\log n$ and matches its complexity in terms of $1/\epsilon$. Technically, our main contribution is the idea of replacing the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the improvement in the quantum query complexity with $\log n$ factors for escaping from saddle points. We also show how to use a quantum gradient computation algorithm due to Jordan to replace the classical gradient queries by quantum evaluation queries with the same complexity. Finally, we also perform numerical experiments that support our theoretical findings.

We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given an $n$-dimensional objective function $f$, our quantum algorithm finds a local minimum using $\tilde{O}(\log^2(n))$ quantum queries to the function value. This quantum algorithm is polynomially better than the classical state-of-the-art algorithm by Jin et al., which requires $\tilde{O}(\log^6(n))$ classical queries to the function gradient. Classical algorithms use perturbations in gradient descent methods to kick the solution path away from saddle points, while we replace the perturbation with a quantum simulation subroutine. Such replacement is possible because we find a quantum particle that escapes from saddle points of its confining potential function. Besides, we also show how to incorporate Jordan's quantum gradient computation routine in our quantum algorithm to remove the gradient oracle assumption in the classical algorithm. Finally, we provide numerical evidence that supports our theoretical findings.

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