Anticoncentration of Complex Gaussian Hafnians

Abstract

Let $G_{2n}$ be a complex symmetric random matrix whose entries above the diagonal are independent standard circular complex Gaussians, and let $H_n=\mathrm{haf}(G_{2n})$. We prove the uniform shifted anticoncentration bound $$\mathrm{Pr}\left(\left|\frac{H_n}{\sqrt{(2n-1)!!}}-z\right|\le \epsilon \right)\le 2\sqrt{\frac{n}{\pi }}{\epsilon }^2$$ for every $z\in \mathbb{C}$ and $\epsilon >0$. This establishes a local anticoncentration property that supports hardness arguments for quantum advantage in Gaussian boson sampling.