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 non-negative 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.

中文翻译：

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非负矩阵哈夫尼随机估计量的性能

高斯玻色子采样器旨在通过执行被认为是经典困难的采样任务来展示量子优势。抽样实验中个体结果的概率由适当构造的对称矩阵的哈夫尼矩阵决定。对于非负矩阵，存在一系列基于生成特定随机矩阵并计算其行列式的哈夫尼随机估计量。虽然这些估计量是无偏的（行列式的均值等于感兴趣的哈夫尼式），但它们的方差可能很高，以至于无法进行有效的估计。在这里，我们研究了两个此类估计器的性能，我们将其称为 Barvinok 和 Godsil-Gutman 估计器。我们发现，一般来说，两种估计量对于随机图的邻接矩阵都表现良好，表明方差随着问题的大小而缓慢增长。尽管如此，在一些简单的例子中，两个估计量都表现出高方差，需要指数数量的样本。此外，我们还计算完整图的方差的渐近行为。最后，我们使用 Godsil-Gutman 估计器模拟高斯玻色子采样，并表明该技术可以成功地再现低阶相关函数。