Proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezed input modes

2026-08-19

AI Summary

该研究证明了具有任意数量压缩输入模式的高斯玻色采样的隐藏猜想，通过特定矩阵近似方法解决了其经典难度论证的一部分。

Abstract

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.

Problem

• Gaussian boson sampling
• hiding conjecture
• classical hardness

Innovation

• Gaussian Boson Sampling
• Hiding Conjecture
• Classical Hardness
• Approximate Instance Generating