We propose quantum algorithms that provide provable speedups for Markov Chain Monte Carlo (MCMC) methods commonly used for sampling from probability distributions of the form $\pi \propto e^{-f}$, where $f$ is a potential function. Our first approach considers Gibbs sampling for finite-sum potentials in the stochastic setting, employing an oracle that provides gradients of individual functions. In the second setting, we consider access only to a stochastic evaluation oracle, allowing simultaneous queries at two points of the potential function under the same stochastic parameter. By introducing novel techniques for stochastic gradient estimation, our algorithms improve the gradient and evaluation complexities of classical samplers, such as Hamiltonian Monte Carlo (HMC) and Langevin Monte Carlo (LMC) in terms of dimension, precision, and other problem-dependent parameters. Furthermore, we achieve quantum speedups in optimization, particularly for minimizing non-smooth and approximately convex functions that commonly appear in empirical risk minimization problems.

翻译：我们提出了量子算法，为马尔可夫链蒙特卡洛（MCMC）方法提供了可证明的加速，这些方法通常用于从形式为 $\pi \propto e^{-f}$ 的概率分布中采样，其中 $f$ 是一个势函数。我们的第一种方法考虑了随机设置下有限和势函数的吉布斯采样，采用了一个提供单个函数梯度的预言机。在第二种设置中，我们仅考虑访问一个随机评估预言机，该预言机允许在相同随机参数下同时查询势函数的两个点。通过引入新颖的随机梯度估计技术，我们的算法在维度、精度和其他问题相关参数方面，改进了经典采样器（如哈密顿蒙特卡洛（HMC）和朗之万蒙特卡洛（LMC））的梯度和评估复杂度。此外，我们在优化问题上实现了量子加速，特别是在最小化经验风险最小化问题中常见的非光滑且近似凸函数方面。