Exact Moments of Gaussian Gram Hafnians Reveal an $n^2/\log n$ Threshold for Weak Anticoncentration
Hongru Zhao
quant-ph
Aug 17, 2026 · v1
math.PR
TL;DR
All sixty-four numbered mathematical equations in the Letter and Supplemental Material were formalized in Lean with kernel checks and axiom reports.
Abstract
Anticoncentration is central to hardness arguments for approximate sampling. In the independent Gaussian surrogate for collision free Gaussian boson sampling, the moment ratio studied here also determines the averaged ideal linear cross entropy reference value. Let $H_{k,n}=\mathrm{haf}(X^{\mathsf T}X)$, where $X\in\mathbb{C}^{k\times 2n}$ has independent standard circular complex Gaussian entries. We evaluate $\mathbb{E}|H_{k,n}|^2$ and $\mathbb{E}|H_{k,n}|^4$ exactly by reducing four hafnian copies to a rank two Gaussian integral. For $R_{k,n}=(\mathbb{E}|H_{k,n}|^2)^2/\mathbb{E}|H_{k,n}|^4$, we obtain $R_{k,n}=4^{-n}\binom{2n}{n}/F_{k,n}$, where $F_{k,n}={}3F_2(-n,-n,1/2;1,k/2;1)$ is a terminating generalized hypergeometric polynomial. If $k/n^2\to c>0$, then $F{k,n}\to e^{1/c}I_0(1/c)$, where $I_0$ is the modified Bessel function of the first kind of order zero, and consequently $R_{k,n}\sqrt{πn}\to[e^{1/c}I_0(1/c)]^{-1}$. Thus $k\asymp n^2$ is a smooth Bessel crossover, whereas the scaling order boundary for inverse polynomial weak anticoncentration is $k\asymp n^2/\log n$. These conclusions concern the Gaussian surrogate moment criterion; finite dimensional Haar moment transfer and high probability small ball anticoncentration remain separate problems.
Problem
Anticoncentration underpins hardness arguments for approximate sampling, and in a Gaussian surrogate for collision-free Gaussian boson sampling a moment ratio determines the averaged ideal linear cross-entropy reference value. The goal is to characterize the scaling regime where weak anticoncentration holds.
Approach
For Gram hafnians H_{k,n}=haf(X^T X) with complex Gaussian X, the second and fourth absolute moments are computed exactly by reducing four hafnian copies to a rank-two Gaussian integral via a Wick auxiliary-field identity. The resulting moment ratio is expressed through a terminating generalized hypergeometric polynomial 3F2. Asymptotics under k/n^2->c are derived using modified Bessel functions. All sixty-four numbered equations were formalized in Lean and checked by the kernel.
Results
The moment ratio equals 4^{-n}C(2n,n)/F_{k,n} with F_{k,n}=3F2(-n,-n,1/2;1,k/2;1). When k/n^2->c>0, F_{k,n}->e^{1/c}I_0(1/c), giving a smooth Bessel crossover at k n^2, while the boundary for inverse-polynomial weak anticoncentration is k n^2/log n.
