Learning Relationship between Quantum Walks and Underdamped Langevin Dynamics

TL;DR

This paper explores the theoretical relationship between quantum walks and underdamped Langevin dynamics, revealing conditions under which they are asymptotically equivalent and discussing implications for machine learning algorithms.

Contribution

It establishes the asymptotic equivalence of randomized quantum walks and underdamped Langevin dynamics, providing new insights into quantum speedup and classical acceleration mechanisms.

Findings

Quantum walk with randomization is asymptotically equivalent to underdamped Langevin dynamics.

Quantum walk without randomization is not asymptotically equivalent due to oscillatory behavior.

Results have implications for the computational and inferential properties of algorithms in machine learning.

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.