Quantum Hamiltonian Descent for Non-smooth Optimization
Jiaqi Leng, Yufan Zheng, Zhiyuan Jia, Lei Fan, Chaoyue Zhao, Yuxiang Peng, Xiaodi Wu
TL;DR
The paper addresses the challenge of non-smooth optimization by introducing Quantum Hamiltonian Descent (QHD), a quantum-dynamics framework that encodes optimization into low-energy quantum states. It develops three continuous-time variants (QHD-SC, QHD-C, QHD-NC) to cover strongly convex, convex, and non-convex non-smooth objectives, establishing global convergence via Lyapunov-energy analyses and, for the non-convex case, finite-time convergence under mild Lipschitz conditions. A fully digitized, discrete-time QHD is then proposed using operator splitting, with convergence properties mirroring the continuous-time results in practice, and complexity analyzed via quantum simulation techniques (including spectral methods and QFT-based diagonalization). Comprehensive numerical experiments show QHD often surpasses classical subgradient methods in non-smooth non-convex settings, highlighting potential quantum advantages in high-dimensional, non-smooth optimization tasks, while also outlining open theoretical and practical challenges for scaling and rigorous discrete-time guarantees.
Abstract
Non-smooth optimization models play a fundamental role in various disciplines, including engineering, science, management, and finance. However, classical algorithms for solving such models often struggle with convergence speed, scalability, and parameter tuning, particularly in high-dimensional and non-convex settings. In this paper, we explore how quantum mechanics can be leveraged to overcome these limitations. Specifically, we investigate the theoretical properties of the Quantum Hamiltonian Descent (QHD) algorithm for non-smooth optimization in both continuous and discrete time. First, we propose continuous-time variants of the general QHD algorithm and establish their global convergence and convergence rate for non-smooth convex and strongly convex problems through a novel Lyapunov function design. Furthermore, we prove the finite-time global convergence of continuous-time QHD for non-smooth non-convex problems under mild conditions (i.e., locally Lipschitz). In addition, we propose discrete-time QHD, a fully digitized implementation of QHD via operator splitting (i.e., product formula). We find that discrete-time QHD exhibits similar convergence properties even with large time steps. Finally, numerical experiments validate our theoretical findings and demonstrate the computational advantages of QHD over classical non-smooth non-convex optimization algorithms.