Year: 2026
Type: preprint
Abstract: Anticoncentration is a key ingredient in complexity theoretic arguments for quantum advantage from Gaussian boson sampling. Here we prove polynomial small ball anticoncentration for complex Gaussian transpose Gram hafnians and complex symmetric Gaussian hafnians. For a k × 2n matrix X with independent standard complex Gaussian entries, we establish polynomial small ball anticoncentration for haf(XᵀX) under k ≥ 4n and n²/k = O(log n). For hafnians of 2n × 2n complex symmetric Gaussian matrices, polynomial small ball anticoncentration holds for every n ≥ 1. In both ensembles, any prescribed inverse polynomial failure probability can be achieved with an inverse polynomial small ball radius, uniformly over all centers. These results go beyond moment based anticoncentration by controlling the probability of small hafnian amplitudes. The proof combines Fourier compression with conditional Wishart geometry to handle dependent hafnian cofactors. Files: main.pdf contains the main article (14 pages); supplement.pdf contains the supplementary proofs and additional results (33 pages). Related Lean verification sources: https://doi.org/10.5281/zenodo.22554594 and https://github.com/HongruZhao/ComplexGramHafnians.
Source: arXiv (Cornell University)
Author: Zhao, Hongru
Institution: University of Minnesota
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213743193