The hiding conjecture for Gaussian boson sampling

Let $U_{NK}$ be the top-left $N\times K$ submatrix of an $M\times M$ Haar-random unitary matrix. Let $Z=Z_{N,K}$ be a random matrix distributed according to either $\mathcal{G}_N^\mathrm{sym}$, the ensemble of symmetric $N\times N$ matrices with independent entries modulo symmetry, diagonal entries distributed as $\mathcal{CN}(0,2)$, and off-diagonal entries as $\mathcal{CN}(0,1)$, or $\mathcal{G}\mathcal{G}^T_{NK}$, the ensemble of matrices $GG^T$ for an $N\times K$ matrix $G$ with independent $\mathcal{CN}(0,1/\sqrt{K})$ entries. Here $d_{\mathrm{TV}}$ denotes total variation distance. Hiding conjecture for Gaussian boson sampling. For $N\leq K\leq M$ and either distribution of $Z$, there exist polynomials $p$ and $r$ such that, for every $\delta>0$ and $M\geq p(N)/r(\delta)$,

$$d_{\mathrm{TV}}\left(MK^{-1/2}U_{NK}U_{NK}^T,Z\right)=O(\delta).$$

This asserts that the relevant submatrix of a Haar unitary product can be hidden in total variation distance as a complex-Gaussian matrix, which is needed to transfer average-case hafnian hardness to approximate Gaussian boson sampling. The paper presents this formulation as adapted from an earlier conjecture and proves it for the maximal-squeezing regime, while the general statement remains the conjectural framework.