  The Approximate Message Passing (AMP) algorithm has garnered significant
attention in recent years for solving linear inverse problems, particularly in
the field of Bayesian inference for high-dimensional models. In this paper, we
consider sampling from the posterior in the linear inverse problem, with an
i.i.d. random design matrix. We develop a sampling algorithm by integrating the
AMP algorithm and stochastic localization. We give a proof for the convergence
in smoothed KL divergence between the distribution of the samples generated by
our algorithm and the target distribution, whenever the noise variance $\Delta$
is below $\Delta_{\rm AMP}$, which is the computation threshold for mean
estimation introduced in (Barbier et al., 2020).
