We study the problem of estimating the partition function $Z(\beta) = \sum_{x \in \Omega} \exp[-\beta \cdot H(x)]$ of a Gibbs distribution defined by a Hamiltonian $H(\cdot)$. It is well known that the partition function $Z(\beta)$ can be well approximated by the simulated annealing method, assuming a sampling oracle that can generate samples according to the Gibbs distribution of any given inverse temperature $\beta$. This method yields the most efficient reductions from counting to sampling, including: $\bullet$ classic non-adaptive (parallel) algorithms with sub-optimal cost [DFK89; Bez+08]; $\bullet$ adaptive (sequential) algorithms with near-optimal cost [SVV09; Hub15; Kol18; HK23]. In this paper, we give an algorithm that achieves efficiency in both parallelism and total work. Specifically, it provides a reduction from counting to sampling using near-optimal total work and logarithmic depth of computation. Consequently, it gives work-efficient parallel counting algorithms for several important models, including the hardcore and Ising models in the uniqueness regime.

翻译：我们研究估计由哈密顿量$H(\cdot)$定义的吉布斯分布配分函数$Z(\beta) = \sum_{x \in \Omega} \exp[-\beta \cdot H(x)]$的问题。众所周知，在假设存在能够根据任意给定逆温度$\beta$的吉布斯分布生成样本的采样预言机时，模拟退火方法可以很好地逼近配分函数$Z(\beta)$。该方法实现了从计数到采样的最有效归约，包括：$\bullet$ 具有次优代价的经典非自适应（并行）算法[DFK89; Bez+08]；$\bullet$ 具有近乎最优代价的自适应（顺序）算法[SVV09; Hub15; Kol18; HK23]。本文提出一种在并行性和总计算量两方面均实现高效的算法。具体而言，该算法利用近乎最优的总计算量和对数级计算深度，实现了从计数到采样的归约。因此，它为若干重要模型提供了工作高效的并行计数算法，包括唯一性区域内的硬核模型与伊辛模型。