  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.
