Local Anticoncentration for Gaussian Boson Sampling via Conditional Wishart Geometry
Abstract
Local anticoncentration controls the conversion of additive output probability estimates to relative estimates in Gaussian boson sampling. We prove a center-uniform quadratic small-ball bound for hafnians of complex Gaussian transpose Gram matrices for 2n selected modes, with an explicit finite coefficient that is polynomial when the number of Gaussian rows is at least of order n2/log n. Letting the Gaussian row count tend to infinity yields anticoncentration for the independent complex symmetric Gaussian ensemble. For the transpose Gram hafnian bound, the proof compresses dependent hafnian cofactors by Fourier interpolation and controls their self-coupled variance through a conditional Wishart identity that preserves the transpose Gram coordinate. Independent perturbations satisfy related bounds, and a positive mixture density shows that the local quadratic power is exact. Together with the uniform hiding theorem in the companion paper, the result gives local anticoncentration for finite Haar interferometers.
Related
- Anticoncentration and entanglement in Gaussian boson sampling
- Affine Geometry of Gaussian ReLU Networks via Conditional Kac-Rice Formulas
- Anticoncentration of Complex Gaussian Hafnians
- Shifted Anticoncentration for Real Gram Hafnians and Symmetric Gaussian Hafnians
- Gram--Wishart--Stiefel formulation of the N=2, large--d gauge theory in 1D: towards emergent Grassmannian geometry
BibTeX
@misc{zhao2026local,
title = {{Local Anticoncentration for Gaussian Boson Sampling via Conditional Wishart Geometry}},
author = {Hongru Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00112},
eprint = {2610.00112},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00112},
}
RIS
TY - UNPB
TI - Local Anticoncentration for Gaussian Boson Sampling via Conditional Wishart Geometry
AU - Hongru Zhao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00112
UR - https://arxiv.org/abs/2610.00112
AB - Local anticoncentration controls the conversion of additive output probability estimates to relative estimates in Gaussian boson sampling. We prove a center-uniform quadratic small-ball bound for hafnians of complex Gaussian transpose Gram matrices for 2n selected modes, with an explicit finite coefficient that is polynomial when the number of Gaussian rows is at least of order n^2/log n. Letting the Gaussian row count tend to infinity yields anticoncentration for the independent complex symmetric Gaussian ensemble. For the transpose Gram hafnian bound, the proof compresses dependent hafnian cofactors by Fourier interpolation and controls their self-coupled variance through a conditional Wishart identity that preserves the transpose Gram coordinate. Independent perturbations satisfy related bounds, and a positive mixture density shows that the local quadratic power is exact. Together with the uniform hiding theorem in the companion paper, the result gives local anticoncentration for finite Haar interferometers.
ER -