Gaussian Boson Samplers aim to demonstrate quantum advantage by performing a sampling task believed to be classically hard. The probabilities of individual outcomes in the sampling experiment are determined by the Hafnian of an appropriately constructed symmetric matrix. For nonnegative matrices, there is a family of randomized estimators of the Hafnian based on generating a particular random matrix and calculating its determinant. While these estimators are unbiased (the mean of the determinant is equal to the Hafnian of interest), their variance may be so high as to prevent an efficient estimation. Here we investigate the performance of two such estimators, which we call the Barvinok and Godsil-Gutman estimators. We find that in general both estimators perform well for adjacency matrices of random graphs, demonstrating a slow growth of variance with the size of the problem. Nonetheless, there are simple examples where both estimators show high variance, requiring an exponential number of samples. In addition, we calculate the asymptotic behavior of the variance for the complete graph. Finally, we simulate the Gaussian Boson Sampling using the Godsil-Gutman estimator and show that this technique can successfully reproduce low-order correlation functions.

翻译：高斯玻色采样旨在通过执行被认为经典困难的任务来展示量子优势。采样实验中单个结果的概率由适当构造的对称矩阵的哈夫尼亚决定。对于非负矩阵，存在一类基于生成特定随机矩阵并计算其行列式的哈夫尼亚随机估计量。尽管这些估计量是无偏的（行列式的均值等于目标哈夫尼亚），但其方差可能过高，导致无法有效估计。本文研究了两种此类估计量——Barvinok估计量和Godsil-Gutman估计量的性能。我们发现，对于随机图的邻接矩阵，这两种估计量通常表现良好，方差随问题规模增长缓慢。然而，存在简单示例表明两种估计量均呈现高方差，需指数级数量的样本。此外，我们计算了完全图方差的渐近行为。最后，我们使用Godsil-Gutman估计量模拟高斯玻色采样，并证明该技术能成功重现低阶关联函数。