发表机构
École Normale Supérieure Paris-Saclay; Imperial College London（巴黎萨克雷高等师范学院; 伦敦帝国理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限状态空间上的动态薛定谔桥问题，利用马尔可夫结构分解为端点耦合和条件桥，提出基于量子行走的吉布斯采样器，将时间范围依赖从二次降至线性，并将端点耦合的矩阵缩放复杂度从二次降至 $N^{3/2}$，同时扩展至加性运行成本模型。
AI 中文摘要
我们考虑有限状态空间上的动态薛定谔桥问题，并利用其马尔可夫结构将问题分解为端点耦合和一系列条件马尔可夫桥。为了从条件桥中采样，我们开发了一种基于桥路径空间上量子行走的量子吉布斯采样器。在转移概率一致为正的条件下，我们推导出谱隙界，并提供了制备初始态的显式程序。与相应经典吉布斯采样器的更新界相比，我们的量子行走查询复杂度将时间范围 $T$ 的依赖从二次改进为线性。对于端点耦合，我们采用量子框约束牛顿法计算薛定谔势，在固定精度下，将矩阵缩放的复杂度从状态空间大小 $N$ 的二次改进为 $N^{3/2}$。最后，我们证明指数重加权将薛定谔桥公式和吉布斯谱隙估计扩展到具有加性运行成本的模型。
英文摘要
We consider the dynamic Schrödinger bridge problem on a finite state space and exploit its Markov structure to decompose the problem into an endpoint coupling and a collection of conditional Markov bridges. To sample from the conditional bridges, we develop a quantum Gibbs sampler based on quantum walks on the bridge path space. Under uniformly positive transition probabilities, we derive a spectral-gap bound and provide an explicit procedure for preparing the initial state. Compared with the update bounds of the corresponding classical Gibbs sampler, our quantum walk-query complexity improves the dependence on the time horizon $T$ from quadratic to linear. For the endpoint coupling, we adapt a quantum box-constrained Newton method to compute the Schrödinger potentials, improving the matrix-scaling complexity, at fixed accuracy, from quadratic to $N^{3/2}$ in the state-space size $N$. Finally, we show that exponential reweighting extends both the Schrödinger bridge formulation and the Gibbs spectral-gap estimate to models with additive running costs.
Comments59 pages, 1 figure
