Uniform Hiding of Haar Block Transpose Gram Matrices

Abstract

Gaussian boson sampling with equally squeezed active inputs assigns collision free probabilities through a complex symmetric transpose Gram matrix formed from a rectangular block of a Haar interferometer. We prove a finite total variation comparison with the corresponding complex Gaussian transpose Gram law. The error bound is explicit, quadratic in the number of selected output modes, inversely proportional to the interferometer size, and uniform in the number of squeezed active inputs. The proof combines a centered circular orthogonal ensemble score analysis in the dense regime with a rectangular relative entropy bound. This result supplies a random matrix replacement component of Gaussian boson sampling hardness arguments.