Subpolynomial query complexity for well-conditioned log-concave sampling

For every fixed ε > 0, we give a sampling algorithm using at most $C_{\epsilon }{d}^{\epsilon }$ exact first-order queries on every execution for C${}^2$ potentials on ℝ${}^d$ with a known minimizer and Hessian between I${}_d$ and $2I_d$. The output has total-variation distance at most 1/10 from the target Gibbs law. Computation between queries is unrestricted. We also prove an $\Omega (\log d)$ query lower bound for arbitrary randomized adaptive algorithms, determining the optimal dimension exponent to be zero.