Quantum spectral method for gradient and Hessian estimation

TL;DR

This paper introduces quantum algorithms for estimating gradients and Hessians of analytic functions, achieving significant speedups over classical methods, especially for sparse Hessians.

Contribution

It extends quantum gradient and Hessian estimation techniques to analytic functions over complex fields, with improved query complexities and new lower bounds.

Findings

Quantum algorithm estimates gradient with $ ilde{O}(1/\varepsilon)$ queries.

Two quantum Hessian estimation algorithms with query complexities $ ilde{O}(d/\varepsilon)$ and $ ilde{O}(d^{1.5}/\varepsilon)$.

Sparse Hessian case yields even more efficient quantum algorithms with $ ilde{O}(s/\varepsilon)$ and $ ilde{O}(sd/\varepsilon)$ queries.

Abstract

Gradient descent is one of the most basic algorithms for solving continuous optimization problems. In [Jordan, PRL, 95(5):050501, 2005], Jordan proposed the first quantum algorithm for estimating gradients of functions close to linear, with exponential speedup in the black-box model. This quantum algorithm was greatly enhanced and developed by [Gily\'en, Arunachalam, and Wiebe, SODA, pp. 1425-1444, 2019], providing a quantum algorithm with optimal query complexity $\widetilde{\Theta}(\sqrt{d}/\varepsilon)$ for a class of smooth functions of $d$ variables, where $\varepsilon$ is the accuracy. This is quadratically faster than classical algorithms for the same problem. In this work, we continue this research by proposing a new quantum algorithm for another class of functions, namely, analytic functions $f(\boldsymbol{x})$ which are well-defined over the complex field. Given phase oracles to query the real and imaginary parts of $f(\boldsymbol{x})$ respectively, we propose a quantum algorithm that returns an $\varepsilon$-approximation of its gradient with query complexity $\widetilde{O}(1/\varepsilon)$. As an extension, we also propose two quantum algorithms for Hessian estimation, aiming to improve quantum analogs of Newton's method. The two algorithms have query complexity $\widetilde{O}(d/\varepsilon)$ and $\widetilde{O}(d^{1.5}/\varepsilon)$, respectively, under different assumptions. Moreover, if the Hessian is promised to be $s$-sparse, we then have two new quantum algorithms with query complexity $\widetilde{O}(s/\varepsilon)$ and $\widetilde{O}(sd/\varepsilon)$, respectively. We also prove a lower bound of $\widetilde{\Omega}(d)$ for Hessian estimation in the general case.