# Quantum Optimization via Gradient-Based Hamiltonian Descent

arXiv: https://arxiv.org/abs/2505.14670 · HTML: https://mlsanity.com/paper/2505.14670/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 44 references that are in the map.
- First posted on arXiv 20 May 2025.

## Builds on (most cited first, 23 of 23)

- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019)
- [On Learning Rates and Schr"odinger Operators](https://mlsanity.com/paper/2004.06977/) (2020)
- [Towards provably efficient quantum algorithms for large-scale machine-learning models](https://mlsanity.com/paper/2303.03428/) (2023)
- [From Nesterov's Estimate Sequence to Riemannian Acceleration](https://mlsanity.com/paper/2001.08876/) (2020)
- [Quantum algorithms for escaping from saddle points](https://mlsanity.com/paper/2007.10253/) (2020)
- [On the Hyperparameters in Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2108.03947/) (2021)
- [On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks](https://mlsanity.com/paper/2209.14501/) (2022)
- [Quantum simulation of real-space dynamics](https://mlsanity.com/paper/2203.17006/) (2022)
- [Time-dependent Hamiltonian simulation with L1-norm scaling](https://mlsanity.com/paper/1906.07115/) (2019)
- [Quantum Hamiltonian Descent](https://mlsanity.com/paper/2303.01471/) (2023)
- [Quantum Langevin Dynamics for Optimization](https://mlsanity.com/paper/2311.15587/) (2023)
- [Accelerated Optimization With Orthogonality Constraints](https://mlsanity.com/paper/1903.05204/) (2019)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [A quantum central path algorithm for linear optimization](https://mlsanity.com/paper/2311.03977/) (2023)
- [A quantum-classical performance separation in nonconvex optimization](https://mlsanity.com/paper/2311.00811/) (2023)
- [Expanding Hardware-Efficiently Manipulable Hilbert Space via Hamiltonian Embedding](https://mlsanity.com/paper/2401.08550/) (2024)
- [QHDOPT: A Software for Nonlinear Optimization with Quantum Hamiltonian Descent](https://mlsanity.com/paper/2409.03121/) (2024)
- [Quantitative Convergences of Lie Group Momentum Optimizers](https://mlsanity.com/paper/2405.20390/) (2024)
- [Scalable Community Detection Using Quantum Hamiltonian Descent and QUBO Formulation](https://mlsanity.com/paper/2411.14696/) (2024)

## Abstract (arXiv)

With rapid advancements in machine learning, first-order algorithms have emerged as the backbone of modern optimization techniques, owing to their computational efficiency and low memory requirements. Recently, the connection between accelerated gradient methods and damped heavy-ball motion, particularly within the framework of Hamiltonian dynamics, has inspired the development of innovative quantum algorithms for continuous optimization. One such algorithm, Quantum Hamiltonian Descent (QHD), leverages quantum tunneling to escape saddle points and local minima, facilitating the discovery of global solutions in complex optimization landscapes. However, QHD faces several challenges, including slower convergence rates compared to classical gradient methods and limited robustness in highly non-convex problems due to the non-local nature of quantum states. Furthermore, the original QHD formulation primarily relies on function value information, which limits its effectiveness. Inspired by insights from high-resolution differential equations that have elucidated the acceleration mechanisms in classical methods, we propose an enhancement to QHD by incorporating gradient information, leading to what we call gradient-based QHD. Gradient-based QHD achieves faster convergence and significantly increases the likelihood of identifying global solutions. Numerical simulations on challenging problem instances demonstrate that gradient-based QHD outperforms existing quantum and classical methods by at least an order of magnitude.