近日，美国马里兰大学Alexey V. Gorshkov团队研究了高斯玻色子采样中Hafnian的二阶矩。相关论文于2025年4月8日发表在《物理评论A》杂志上。

高斯玻色子采样是实验证明量子优势的一种流行方法，但在充分理解其理论基础方面仍存在许多微妙之处。采样近似平均硬度理论论证中的一个重要组成部分是反集中，这是输出概率的二阶矩性质。在高斯玻色子采样中，这些由广义圆形正交系综矩阵的hafnian给出。在Ehrenberg等人的一项配套工作中，他们开发了一种图论方法来研究这些矩，并用它来识别反浓缩的转变。

在这项工作中，研究组使用这些图论技术找到了二阶矩的递归表达式。虽然他们无法手动求解这个递归，但能够精确地用数值求解它，直到Fock扇区2??=80。研究组进一步推导了关于二阶矩的分析结果。这些结果使人们能够精确定位反浓缩的转变，并进一步得出理想（无误差）设备的预期线性交叉熵基准分数。

附：英文原文

Title: Second moment of Hafnians in Gaussian boson sampling

Author: Adam Ehrenberg, Joseph T. Iosue, Abhinav Deshpande, Dominik Hangleiter, Alexey V. Gorshkov

Issue&Volume: 2025/04/08

Abstract: Gaussian boson sampling is a popular method for experimental demonstrations of quantum advantage, but many subtleties remain in fully understanding its theoretical underpinnings. An important component in the theoretical arguments for approximate average-case hardness of sampling is anticoncentration, which is a second-moment property of the output probabilities. In Gaussian boson sampling these are given by hafnians of generalized circular orthogonal ensemble matrices. In a companion work by Ehrenberg et al. [Phys. Rev. Lett. 134, 140601 (2025)], we develop a graph-theoretic method to study these moments and use it to identify a transition in anticoncentration. In this work, we find a recursive expression for the second moment using these graph-theoretic techniques. While we have not been able to solve this recursion by hand, we are able to solve it numerically exactly, which we do up to Fock sector 2n=80. We further derive analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device.

DOI: 10.1103/PhysRevA.111.042412

Source: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.111.042412