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.

翻译：快速计算算法需求持续增长，其发展受到量子加速与经典加速等进展的推动。本文旨在研究量子计算中基于量子行走的搜索算法，以及经典计算中基于朗之万动力学的采样算法。在量子计算方面，基于量子行走的搜索算法相较于经典算法可实现二次加速。在经典计算领域，大量研究聚焦于梯度加速，其中源自欠阻尼朗之万动力学的梯度调整算法相比传统朗之万算法能提供二次加速。由于搜索算法与采样算法均旨在解决学习任务，我们研究了投币量子行走与欠阻尼朗之万动力学之间的学习关系。具体而言，我们证明：在Le Cam缺陷距离度量下，随机化的量子行走渐近等价于欠阻尼朗之万动力学，而未随机化的量子行走因其高频振荡特性不具备渐近等价性。我们进一步探讨了这些等价性与非等价性结果对机器学习任务中相关算法的计算特性与推断性质的影响。本研究为理解量子行走与欠阻尼朗之万动力学的关系，以及量子加速与经典梯度加速的内在机制提供了新的见解。