Gibbs sampling from continuous real-valued functions is a challenging problem of interest in machine learning. Here we leverage quantum Fourier transforms to build a quantum algorithm for this task when the function is periodic. We use the quantum algorithms for solving linear ordinary differential equations to solve the Fokker--Planck equation and prepare a quantum state encoding the Gibbs distribution. We show that the efficiency of interpolation and differentiation of these functions on a quantum computer depends on the rate of decay of the Fourier coefficients of the Fourier transform of the function. We view this property as a concentration of measure in the Fourier domain, and also provide functional analytic conditions for it. Our algorithm makes zeroeth order queries to a quantum oracle of the function. Despite suffering from an exponentially long mixing time, this algorithm allows for exponentially improved precision in sampling, and polynomial quantum speedups in mean estimation in the general case, and particularly under geometric conditions we identify for the critical points of the energy function.

翻译：吉布斯采样从连续实值函数中采样是机器学习领域一个具有挑战性的问题。本文利用量子傅里叶变换，针对周期性函数构建了相应的量子算法。我们采用求解线性常微分方程的量子算法来求解福克-普朗克方程，并制备编码吉布斯分布的量子态。研究表明，量子计算机上对这些函数进行插值和微分的效率取决于函数傅里叶变换中傅里叶系数的衰减速率。我们将此性质视为傅里叶域中的测度集中现象，并为其提供了泛函分析条件。该算法对函数的量子神谕进行零阶查询。尽管存在指数级长的混合时间，但该算法在采样精度上实现了指数级提升，并且在一般情况下（特别是在我们确定的能量函数临界点几何条件下）实现了均值估计的多项式量子加速。