函数凸性测试的量子算法
Quantum Algorithm For Testing Convexity of Function
AI总结：
本文提出一种测试多项式函数凸性的量子算法，证明其在变量数量上相比经典计算机具有超多项式加速，并改进和扩展了此前的量子牛顿法，可应用于流形几何、变分量子算法训练景观及优化方法研究。
摘要：
函数是数学中的基本对象，在不同领域有无数的应用，并且通常根据其定义域和值域以及某些性质进行分类。实值函数的一个重要性质是其凸性，它在热力学和几何学等许多领域中起着非常关键的作用。受量子计算最新进展以及追求量子优势的推动，我们给出了一种用于测试多项式函数凸性的量子算法，多项式函数在优化、机器学习、物理学等多种情境中频繁出现。我们证明，就变量数量而言，量子计算机揭示凸性性质的速度比经典计算机超多项式地更快。作为推论，我们对 Rebentrost 等人早期工作 [New J. Phys. \ extbf{21} 073023 (2019)] 中构建的量子牛顿法提供了显著的改进和扩展。我们进一步在更广泛的背景下讨论我们的算法，例如在流形几何结构研究中的潜在应用、变分量子算法训练景观的测试，以及用于优化的梯度下降/牛顿法。
英文摘要：
Functions are a fundamental object in mathematics, with countless applications to different fields, and are usually classified based on certain properties, given their domains and images. An important property of a real-valued function is its convexity, which plays a very crucial role in many areas, such as thermodynamics and geometry. Motivated by recent advances in quantum computation as well as the quest for quantum advantage, we give a quantum algorithm for testing convexity of polynomial functions, which appears frequently in multiple contexts, such as optimization, machine learning, physics, etc. We show that quantum computers can reveal the convexity property superpolynomially faster than classical computers with respect to number of variables. As a corollary, we provide a significant improvement and extension on quantum Newton's method constructed in earlier work of Rebentrost et al [New J. Phys. \textbf{21} 073023 (2019)]. We further discuss our algorithm in a broader context, such as potential application in the study of geometric structure of manifold, testing training landscape of variational quantum algorithm and also gradient descent/Newton's method for optimization.
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