Quantum spectral method for gradient and Hessian estimation

• URL: http://arxiv.org/abs/2407.03833v1
• Date: Thu, 4 Jul 2024 11:03:48 GMT
• Title: Quantum spectral method for gradient and Hessian estimation
• Authors: Yuxin Zhang, Changpeng Shao,
• Abstract summary: Gradient descent is one of the most basic algorithms for solving continuous optimization problems. We propose a quantum algorithm that returns an $varepsilon$-approximation of its gradient with query complexity $widetildeO (1/varepsilon)$. We also propose two quantum algorithms for Hessian estimation, aiming to improve quantum analogs of Newton's method.
• Score: 4.193480001271463
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\widetilde{O}(1/\varepsilon)$. This achieves exponential speedup over classical algorithms in terms of the dimension $d$. 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. The former achieves exponential speedup over classical algorithms. We also prove a lower bound of $\widetilde{\Omega}(d)$ for Hessian estimation in the general case.

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