A Faster Approximation Algorithm for the Gibbs Partition Function

We consider the problem of estimating the partition function Z(β) = ∑ x exp(−β(H(x)) of a Gibbs distribution with a Hamilton H(·), or more precisely the logarithm of the ratio q = lnZ(0)/Z(β). 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, β]. The current best known approach due to Huber [9] uses O(q lnn · [ln q+ln lnn+ε]) oracle calls on average where ε is the desired accuracy of approximation and H(·) is assumed to lie in [1, n]. We improve the complexity to O(q lnn · ε) oracle calls.