We present classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Our work has two main contributions: first, we modify the classical algorithm of Stefankovic, Vempala and Vigoda (J. ACM, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020).
The conventional approach to estimating partition functions requires approximating the means of Gibbs distributions at a set of inverse temperatures that form the so-called cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of Stefankovic, Vempala and Vigoda with the paired-product estimator of Huber (Ann. Appl. Probab., 25(2), 2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by Stefankovic, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn
by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.

中文翻译：

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Gibbs 分区函数的更简单（经典）和更快（量子）算法

我们提出了经典和量子算法，用于在给定温度下逼近经典哈密顿量的配分函数。我们的工作有两个主要贡献：首先，我们修改了 Stefankovic、Vempala 和 Vigoda (J. ACM, 56(3), 2009) 的经典算法以提高其样本复杂度；其次，我们量化了这个新算法，改进了之前最快的量子算法，这要归功于 Harrow 和 Wei（SODA 2020）。
估计配分函数的传统方法需要在一组逆温度下近似吉布斯分布的平均值，这些温度形成所谓的冷却计划。冷却计划的长短直接影响算法的复杂性。将我们的 Stefankovic、Vempala 和 Vigoda 算法的改进版本与 Huber 的配对乘积估计器（Ann. Appl. Probab., 25(2), 2015）相结合，我们的新量子算法使用比以前已知的更短的冷却时间表。这个长度与 Stefankovic、Vempala 和 Vigoda 推测的最佳长度相匹配。与抽取的随机样本数量相比，量子算法在所需的量子样本数量方面也实现了二次优势
通过最好的经典算法，其计算复杂度对用于产生量子样本的马尔可夫链的光谱间隙具有二次更好的依赖性。