Quantum Algorithm For Testing Convexity of Function

Abstract

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.