  We consider the problem of estimating the partition function $Z(\beta)=\sum_x
\exp(-\beta(H(x))$ of a Gibbs distribution with a Hamilton $H(\cdot)$, or more
precisely the logarithm of the ratio $q=\ln Z(0)/Z(\beta)$. It has been
recently shown how to approximate $q$ with high probability assuming the
existence of an oracle that produces samples from the Gibbs distribution for a
given parameter value in $[0,\beta]$. The current best known approach due to
Huber [9] uses $O(q\ln n\cdot[\ln q + \ln \ln n+\varepsilon^{-2}])$ oracle
calls on average where $\varepsilon$ is the desired accuracy of approximation
and $H(\cdot)$ is assumed to lie in $\{0\}\cup[1,n]$. We improve the complexity
to $O(q\ln n\cdot\varepsilon^{-2})$ oracle calls. We also show that the same
complexity can be achieved if exact oracles are replaced with approximate
sampling oracles that are within $O(\frac{\varepsilon^2}{q\ln n})$ variation
distance from exact oracles. Finally, we prove a lower bound of $\Omega(q\cdot
\varepsilon^{-2})$ oracle calls under a natural model of computation.
