# Quantum Hamiltonian Descent

arXiv: https://arxiv.org/abs/2303.01471 · HTML: https://mlsanity.com/paper/2303.01471/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 44 references that are in the map.
- First posted on arXiv 2 Mar 2023.

## Builds on (most cited first, 5 of 5)

- [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)
- [A Literature Survey of Benchmark Functions For Global Optimization Problems](https://mlsanity.com/paper/1308.4008/) (2013)
- [Adiabatic Quantum Linear Regression](https://mlsanity.com/paper/2008.02355/) (2020)
- [Quantum simulation of real-space dynamics](https://mlsanity.com/paper/2203.17006/) (2022)

## Built on by (most cited first, 7 of 7)

- [Quantum Langevin Dynamics for Optimization](https://mlsanity.com/paper/2311.15587/) (2023)
- [Quantum computing and artificial intelligence: status and perspectives](https://mlsanity.com/paper/2505.23860/) (2025)
- [Quantum Non-Linear Bandit Optimization](https://mlsanity.com/paper/2503.03023/) (2025)
- [Quantum Algorithms for Bandits with Knapsacks with Improved Regret and Time Complexities](https://mlsanity.com/paper/2507.04438/) (2025)
- [Scalable Community Detection Using Quantum Hamiltonian Descent and QUBO Formulation](https://mlsanity.com/paper/2411.14696/) (2024)
- [Operator-Level Quantum Acceleration of Non-Logconcave Sampling](https://mlsanity.com/paper/2505.05301/) (2025)
- [Quantum Optimization via Gradient-Based Hamiltonian Descent](https://mlsanity.com/paper/2505.14670/) (2025)

## Abstract (arXiv)

Gradient descent is a fundamental algorithm in both theory and practice for continuous optimization. Identifying its quantum counterpart would be appealing to both theoretical and practical quantum applications. A conventional approach to quantum speedups in optimization relies on the quantum acceleration of intermediate steps of classical algorithms, while keeping the overall algorithmic trajectory and solution quality unchanged. We propose Quantum Hamiltonian Descent (QHD), which is derived from the path integral of dynamical systems referring to the continuous-time limit of classical gradient descent algorithms, as a truly quantum counterpart of classical gradient methods where the contribution from classically-prohibited trajectories can significantly boost QHD's performance for non-convex optimization. Moreover, QHD is described as a Hamiltonian evolution efficiently simulatable on both digital and analog quantum computers. By embedding the dynamics of QHD into the evolution of the so-called Quantum Ising Machine (including D-Wave and others), we empirically observe that the D-Wave-implemented QHD outperforms a selection of state-of-the-art gradient-based classical solvers and the standard quantum adiabatic algorithm, based on the time-to-solution metric, on non-convex constrained quadratic programming instances up to 75 dimensions. Finally, we propose a "three-phase picture" to explain the behavior of QHD, especially its difference from the quantum adiabatic algorithm.