We investigate the performance of two approximation algorithms for the Hafnian of a nonnegative square matrix, namely the Barvinok and Godsil-Gutman estimators. We observe that, while there are examples of matrices for which these algorithms fail to provide a good approximation, the algorithms perform surprisingly well for adjacency matrices of random graphs. In most cases, the Godsil-Gutman estimator provides a far superior accuracy. For dense graphs, however, both estimators demonstrate a slow growth of the variance. For complete graphs, we show analytically that the relative variance $\sigma / \mu$ grows as a square root of the size of the graph. Finally, we simulate a Gaussian Boson Sampling experiment using the Godsil-Gutman estimator and show that the technique used can successfully reproduce low-order correlation functions.

翻译：本文研究了两种用于非负平方矩阵Hafnian近似估计算法的性能，即Barvinok估计量和Godsil-Gutman估计量。我们观察到，尽管存在这些算法无法提供良好近似的矩阵实例，但对于随机图的邻接矩阵，这两种算法的表现却出人意料地优异。在多数情况下，Godsil-Gutman估计量能够提供远胜于Barvinok估计量的精度。然而，对于稠密图，两种估计量的方差均呈现缓慢增长态势。针对完全图，我们通过解析推导证明相对方差 $\sigma / \mu$ 随图的规模平方根增长。最后，我们利用Godsil-Gutman估计量模拟高斯玻色采样实验，结果表明该技术能成功复现低阶关联函数。