  We address the outstanding problem of sampling from an unnormalized density
that may be non-log-concave and multimodal. To enhance the performance of
simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type
have been widely used. However, quantitative theoretical guarantees of these
techniques are under-explored. This study takes a first step toward providing a
non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the
first time, an oracle complexity of $\widetilde{O}\left(\frac{d\beta^2{\cal
A}^2}{\varepsilon^6}\right)$ for simple annealed Langevin Monte Carlo algorithm
to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the
target distribution $\pi\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with
$\beta$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve
of probability measures interpolating the target distribution $\pi$ and a
readily sampleable distribution.
