  We consider the problem of algorithmically sampling from the Gibbs measure of
a mixed $p$-spin spherical spin glass. We give a polynomial-time algorithm that
samples from the Gibbs measure up to vanishing total variation error, for any
model whose mixture satisfies $$\xi''(s) < \frac{1}{(1-s)^2}, \qquad \forall
s\in [0,1).$$ This includes the pure $p$-spin glasses above a critical
temperature that is within an absolute ($p$-independent) constant of the
so-called shattering phase transition. Our algorithm follows the algorithmic
stochastic localization approach introduced in (Alaoui, Montanari, Sellke,
20022). A key step of this approach is to estimate the mean of a sequence of
tilted measures. We produce an improved estimator for this task by identifying
a suitable correction to the TAP fixed point selected by approximate message
passing (AMP). As a consequence, we improve the algorithm's guarantee over
previous work, from normalized Wasserstein to total variation error. In
particular, the new algorithm and analysis opens the way to perform inference
about one-dimensional projections of the measure.
