Motivated by applications in machine learning, we present a quantum algorithm for Gibbs sampling from continuous real-valued functions defined on high dimensional tori. We show that these families of functions satisfy a Poincar\'e inequality. We then use the techniques for solving linear systems and partial differential equations to design an algorithm that performs zeroeth order queries to a quantum oracle computing the energy function to return samples from its Gibbs distribution. We then analyze the query and gate complexity of our algorithm and prove that the algorithm has a polylogarithmic dependence on approximation error (in total variation distance) and a polynomial dependence on the number of variables, although it suffers from an exponentially poor dependence on temperature. To this end, we develop provably efficient quantum algorithms for manipulating real analytic periodic functions.

翻译：受机器学习应用的驱动，我们提出了一种量子算法，用于从定义在高维环面上的连续实值函数中进行Gibbs采样。我们证明了这些函数族满足Poincaré不等式。随后，利用求解线性系统和偏微分方程的技术，我们设计了一种算法，该算法通过对计算能量函数的量子预言机进行零阶查询，从而返回其Gibbs分布的样本。接着，我们分析了该算法的查询复杂度和门复杂度，并证明该算法对近似误差（以总变差距离衡量）具有多对数依赖关系，对变量数量具有多项式依赖关系，尽管其对温度的依赖关系呈指数级恶化。为此，我们开发了可证明高效的量子算法，用于处理实解析周期函数。