Learning Relationship between Quantum Walks and Underdamped Langevin Dynamics
Yazhen Wang
TL;DR
This paper investigates the learning relationship between quantum walk–based search and underdamped Langevin dynamics used for accelerated sampling. By framing both paradigms as learning problems and employing Le Cam deficiency distance, it proves that randomized quantum walks are asymptotically equivalent to underdamped Langevin dynamics for inference, while nonrandomized quantum walks are not due to high-frequency oscillations. The work highlights a shared lifting mechanism driving quadratic speedups in both quantum and classical settings and explains why randomization is essential for cross-model equivalence. These results provide a rigorous bridge between quantum speedup and classical gradient acceleration, with potential implications for designing cross-domain algorithms and understanding the limits of acceleration in ML tasks.
Abstract
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.