Anticoncentration of Complex Gaussian Hafnians

Source abstract

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