任意压缩输入模式数的高斯玻色采样的隐藏猜想证明
Proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezed input modes
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中文总结 AI 辅助
该研究完成了任意压缩输入模式数的高斯玻色采样的隐藏猜想证明，发现原有基于密度的实例生成方法在K=cM且c<1/2时失效，转而采用近似实例生成实现经典难度归约。
中文摘要 AI 辅助
高斯玻色采样（GBS）是一项为展示量子优势而提出的采样任务。我们考虑具有M个光学模式、K个等压缩输入模式和N个观测光子计数的高斯玻色采样。我们完成了具有任意数量压缩器K的高斯玻色采样的隐藏猜想证明，这是GBS经典难度论证的一部分。特别地，我们证明对于任意K和N=o(√K)，对称乘积MK^(-1/2)U_{NK}U_{NK}^T（其中U_{NK}是M×M哈尔随机幺正矩阵U的左上角N×K子矩阵），在总变差距离上既接近N×N对称复高斯矩阵G（其元素独立），又接近对称乘积GG^T/√K（其中G是N×K个独立同分布标准复高斯矩阵）。然而，我们发现[Aaronson和Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8]中用于高效实现隐藏过程的基于密度的实例生成方法，在K=cM且c<1/2时，对高斯玻色采样失效。相反，我们使用近似实例生成来为常规经典难度归约实现隐藏。
英文摘要
Gaussian boson sampling (GBS) is a sampling task proposed to demonstrate quantum advantage. We consider Gaussian boson sampling on $M$ optical modes, with $K$ equally squeezed input modes and $N$ observed photon counts. We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers $K$, which is a part of the argument for classical hardness of GBS. In particular, we show that for any $K$ and $N=o(\sqrt{K})$, the symmetric product $MK^{-1/2}U_{NK}U_{NK}^T$, for $U_{NK}$ the top left $N\times K$ submatrix of an $M\times M$ Haar random unitary $U$, is close in total variation distance to both an $N\times N$ symmetric complex Gaussian matrix $\mathbf G$ with independent entries, and the symmetric product $GG^T/\sqrt{K}$ for $G$ an $N\times K$ matrix of iid standard complex Gaussians. We show however that the density-based instance generating method of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8] used to efficiently implement a hiding procedure fails for Gaussian boson sampling with $K=cM$ if $c<1/2$. Instead we use approximate instance generating to implement the hiding for the usual classical hardness reduction.
