# Quantum Langevin Dynamics for Optimization

arXiv: https://arxiv.org/abs/2311.15587 · HTML: https://mlsanity.com/paper/2311.15587/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 83 references that are in the map.
- First posted on arXiv 27 Nov 2023.

## Builds on (most cited first, 28 of 28)

- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Non-convex Optimization for Machine Learning](https://mlsanity.com/paper/1712.07897/) (2017)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [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)
- [A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1702.05575/) (2017)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [On Learning Rates and Schr"odinger Operators](https://mlsanity.com/paper/2004.06977/) (2020)
- [Quantum SDP-Solvers: Better upper and lower bounds](https://mlsanity.com/paper/1705.01843/) (2017)
- [Convex optimization using quantum oracles](https://mlsanity.com/paper/1809.00643/) (2018)
- [Sublinear quantum algorithms for training linear and kernel-based classifiers](https://mlsanity.com/paper/1904.02276/) (2019)
- [A Quantum Interior Point Method for LPs and SDPs](https://mlsanity.com/paper/1808.09266/) (2018)
- [Adiabatic Quantum Linear Regression](https://mlsanity.com/paper/2008.02355/) (2020)
- [Quantum algorithms and lower bounds for convex optimization](https://mlsanity.com/paper/1809.01731/) (2018)
- [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)
- [Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants](https://mlsanity.com/paper/2210.06539/) (2022)
- [Time-dependent Hamiltonian simulation with L1-norm scaling](https://mlsanity.com/paper/1906.07115/) (2019)
- [Quantum Speedups for Zero-Sum Games via Improved Dynamic Gibbs Sampling](https://mlsanity.com/paper/2301.03763/) (2023)
- [Quantum Hamiltonian Descent](https://mlsanity.com/paper/2303.01471/) (2023)
- [Quantum Interior Point Methods for Semidefinite Optimization](https://mlsanity.com/paper/2112.06025/) (2021)
- [Logarithmic-Regret Quantum Learning Algorithms for Zero-Sum Games](https://mlsanity.com/paper/2304.14197/) (2023)
- [New hard benchmark functions for global optimization](https://mlsanity.com/paper/2202.04606/) (2022)
- [A quantum-classical performance separation in nonconvex optimization](https://mlsanity.com/paper/2311.00811/) (2023)

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

- [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)
- [Operator-Level Quantum Acceleration of Non-Logconcave Sampling](https://mlsanity.com/paper/2505.05301/) (2025)
- [A Spin Glass Characterization of Neural Networks](https://mlsanity.com/paper/2508.07397/) (2025)
- [Gradient Testing and Estimation by Comparisons](https://mlsanity.com/paper/2405.11454/) (2024)
- [Quantum Optimization via Gradient-Based Hamiltonian Descent](https://mlsanity.com/paper/2505.14670/) (2025)
- [Quantum Algorithms for Projection-Free Sparse Convex Optimization](https://mlsanity.com/paper/2507.08543/) (2025)

## Abstract (arXiv)

We initiate the study of utilizing Quantum Langevin Dynamics (QLD) to solve optimization problems, particularly those non-convex objective functions that present substantial obstacles for traditional gradient descent algorithms. Specifically, we examine the dynamics of a system coupled with an infinite heat bath. This interaction induces both random quantum noise and a deterministic damping effect to the system, which nudge the system towards a steady state that hovers near the global minimum of objective functions. We theoretically prove the convergence of QLD in convex landscapes, demonstrating that the average energy of the system can approach zero in the low temperature limit with an exponential decay rate correlated with the evolution time. Numerically, we first show the energy dissipation capability of QLD by retracing its origins to spontaneous emission. Furthermore, we conduct detailed discussion of the impact of each parameter. Finally, based on the observations when comparing QLD with classical Fokker-Plank-Smoluchowski equation, we propose a time-dependent QLD by making temperature and hbar time-dependent parameters, which can be theoretically proven to converge better than the time-independent case and also outperforms a series of state-of-the-art quantum and classical optimization algorithms in many non-convex landscapes.