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

arXiv:2608.19314v1 Announce Type: new 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.

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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Mathematical Physics (math-ph)

arXiv:2608.19314 [quant-ph]

(or arXiv:2608.19314v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.19314 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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

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