# Learning Relationship between Quantum Walks and Underdamped Langevin Dynamics

arXiv: https://arxiv.org/abs/2601.01589 · HTML: https://mlsanity.com/paper/2601.01589/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 59 references that are in the map.
- First posted on arXiv 4 Jan 2026.

## Builds on (most cited first, 13 of 13)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Variational Wasserstein gradient flow](https://mlsanity.com/paper/2112.02424/) (2021)
- [Generative quantum machine learning via denoising diffusion probabilistic models](https://mlsanity.com/paper/2310.05866/) (2023)
- [Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2301.01766/) (2023)
- [Asymptotic Analysis via Stochastic Differential Equations of Gradient Descent Algorithms in Statistical and Computational Paradigms](https://mlsanity.com/paper/1711.09514/) (2017)
- [Lifting Markov Chains To Mix Faster: Limits and Opportunities](https://mlsanity.com/paper/1705.08253/) (2017)

## Abstract (arXiv)

Fast computational algorithms are in constant demand, and their development has been driven by advances such as quantum speedup and classical acceleration. This paper intends to study search algorithms based on quantum walks in quantum computation and sampling algorithms based on Langevin dynamics in classical computation. On the quantum side, quantum walk-based search algorithms can achieve quadratic speedups over their classical counterparts. In classical computation, a substantial body of work has focused on gradient acceleration, with gradient-adjusted algorithms derived from underdamped Langevin dynamics providing quadratic acceleration over conventional Langevin algorithms. Since both search and sampling algorithms are designed to address learning tasks, we study learning relationship between coined quantum walks and underdamped Langevin dynamics. Specifically, we show that, in terms of the Le Cam deficiency distance, a quantum walk with randomization is asymptotically equivalent to underdamped Langevin dynamics, whereas the quantum walk without randomization is not asymptotically equivalent due to its high-frequency oscillatory behavior. We further discuss the implications of these equivalence and nonequivalence results for the computational and inferential properties of the associated algorithms in machine learning tasks. Our findings offer new insight into the relationship between quantum walks and underdamped Langevin dynamics, as well as the intrinsic mechanisms underlying quantum speedup and classical gradient acceleration.