  Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, the problem of
sampling from a distribution $\propto e^{-f(x)}$ is called log-concave
sampling. This task has wide applications in machine learning, physics,
statistics, etc. In this work, we develop quantum algorithms for sampling
log-concave distributions and for estimating their normalizing constants
$\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x$. First, we use underdamped Langevin
diffusion to develop quantum algorithms that match the query complexity (in
terms of the condition number $\kappa$ and dimension $d$) of analogous
classical algorithms that use gradient (first-order) queries, even though the
quantum algorithms use only evaluation (zeroth-order) queries. For estimating
normalizing constants, these algorithms also achieve quadratic speedup in the
multiplicative error $\epsilon$. Second, we develop quantum Metropolis-adjusted
Langevin algorithms with query complexity $\widetilde{O}(\kappa^{1/2}d)$ and
$\widetilde{O}(\kappa^{1/2}d^{3/2}/\epsilon)$ for log-concave sampling and
normalizing constant estimation, respectively, achieving polynomial speedups in
$\kappa,d,\epsilon$ over the best known classical algorithms by exploiting
quantum analogs of the Monte Carlo method and quantum walks. We also prove a
$1/\epsilon^{1-o(1)}$ quantum lower bound for estimating normalizing constants,
implying near-optimality of our quantum algorithms in $\epsilon$.
