  We derive explicit bounds for the computation of normalizing constants $Z$
for log-concave densities $\pi = \exp(-U)/Z$ with respect to the Lebesgue
measure on $\mathbb{R}^d$. Our approach relies on a Gaussian annealing combined
with recent and precise bounds on the Unadjusted Langevin Algorithm
(High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm, A.
Durmus and E. Moulines). Polynomial bounds in the dimension $d$ are obtained
with an exponent that depends on the assumptions made on $U$. The algorithm
also provides a theoretically grounded choice of the annealing sequence of
variances. A numerical experiment supports our findings. Results of independent
interest on the mean squared error of the empirical average of locally
Lipschitz functions are established.
