arXiv:2609.35019

Anticoncentration of Complex Gaussian Hafnians

Priyanshu Pant

Published Sep 28, 2026• 0 citations

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.

DOI: 10.48550/arxiv.2609.35019

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arXiv:2609.35019

Anticoncentration of Complex Gaussian Hafnians

Priyanshu Pant

Published Sep 28, 2026• 0 citations

arXiv HTML PDF

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.

DOI: 10.48550/arxiv.2609.35019

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Local Anticoncentration for Gaussian Boson Sampling via Conditional Wishart Geometry

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Gaussian Boson Sampling is a popular method for experimental demonstrations of quantum advantage, but many subtleties remain in fully understanding its theoretical underpinnings. An important component in the theoretical arguments for approximate average-case hardness of sampling is anticoncentration, which is a second-moment property of the output probabilities. In Gaussian Boson Sampling these are given by hafnians of generalized circular orthogonal ensemble matrices. In a companion work [arXiv:2312.08433], we develop a graph-theoretic method to study these moments and use it to identify a transition in anticoncentration. In this work, we find a recursive expression for the second moment using these graph-theoretic techniques. While we have not been able to solve this recursion by hand, we are able to solve it numerically exactly, which we do up to Fock sector $2n = 80$ . We further derive new analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device.Gaussian Boson Sampling is a popular method for experimental demonstrations of quantum advantage, but many subtleties remain in fully understanding its theoretical underpinnings. An important component in the theoretical arguments for approximate average-case hardness of sampling is anticoncentration, which is a second-moment property of the output probabilities. In Gaussian Boson Sampling these are given by hafnians of generalized circular orthogonal ensemble matrices. In a companion work [arXiv:2312.08433], we develop a graph-theoretic method to study these moments and use it to identify a transition in anticoncentration. In this work, we find a recursive expression for the second moment using these graph-theoretic techniques. While we have not been able to solve this recursion by hand, we are able to solve it numerically exactly, which we do up to Fock sector $2n = 80$. We further derive new analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device.

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Anticoncentration of the Permanent in Ginibre Ensembles

Frederic Koehler, Pui Kuen Leung

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$ , put $β=\dim_{\mathbb{R}}\mathbb{K}$ , and let $G_n^{\mathbb{K}}$ be an $n\times n$ matrix with i.i.d. standard $\mathbb{K}$ -Gaussian entries, namely a standard $\mathbb{K}$ -Ginibre matrix. We prove that the normalized row-ordered permanent $W_n^{\mathbb{K}}=\operatorname{per}_{\mathbb{K}}G_n^{\mathbb{K}}/\sqrt{n!}$ has a radial density $p_n^{\mathbb{K}}$ satisfying $\|p_n^{\mathbb{K}}\|_\infty=p_n^{\mathbb{K}}(0)\lesssim_βn^{(β+2)/4}$ and $\sup_{z\in\mathbb{K}}\mathbb{P}(|W_n^{\mathbb{K}}-z|\leq\varepsilon)\lesssim_βn^{(β+2)/4}\varepsilon^β$ . In particular, for $\mathbb{K}=\mathbb{C}$ , this resolves the Permanent Anticoncentration Conjecture of Aaronson and Arkhipov. The proof compares the squared Gaussian permanent with the squared (Study) determinant in Laplace-transform order.Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, put $β=\dim_{\mathbb{R}}\mathbb{K}$, and let $G_n^{\mathbb{K}}$ be an $n\times n$ matrix with i.i.d. standard $\mathbb{K}$-Gaussian entries, namely a standard $\mathbb{K}$-Ginibre matrix. We prove that the normalized row-ordered permanent $W_n^{\mathbb{K}}=\operatorname{per}_{\mathbb{K}}G_n^{\mathbb{K}}/\sqrt{n!}$ has a radial density $p_n^{\mathbb{K}}$ satisfying $\|p_n^{\mathbb{K}}\|_\infty=p_n^{\mathbb{K}}(0)\lesssim_βn^{(β+2)/4}$ and $\sup_{z\in\mathbb{K}}\mathbb{P}(|W_n^{\mathbb{K}}-z|\leq\varepsilon)\lesssim_βn^{(β+2)/4}\varepsilon^β$. In particular, for $\mathbb{K}=\mathbb{C}$, this resolves the Permanent Anticoncentration Conjecture of Aaronson and Arkhipov. The proof compares the squared Gaussian permanent with the squared (Study) determinant in Laplace-transform order.

math.PRCross-listedarXiv:2607.20329Jul 23, 2026

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.

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.

Exact Moments of Gaussian Gram Hafnians Reveal an

n^2/\log n

Threshold for Weak Anticoncentration

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.

α

Page curves for Gaussian boson sampling. (3) We study unequal input squeezing parameters

(s_i)_i

, and prove estimates as well as monotonicity of the average-case Rényi-2 entropy Page curve in terms of the magnitudes

|s_i|

, which allow for extending equal squeezing results to unequal squeezing.

2n = 80

. We further derive new analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device.

Let

\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}

, put

β=\dim_{\mathbb{R}}\mathbb{K}

, and let

G_n^{\mathbb{K}}

be an

n\times n

matrix with i.i.d. standard

\mathbb{K}

-Gaussian entries, namely a standard

\mathbb{K}

-Ginibre matrix. We prove that the normalized row-ordered permanent

W_n^{\mathbb{K}}=\operatorname{per}_{\mathbb{K}}G_n^{\mathbb{K}}/\sqrt{n!}

has a radial density

p_n^{\mathbb{K}}

satisfying

\|p_n^{\mathbb{K}}\|_\infty=p_n^{\mathbb{K}}(0)\lesssim_βn^{(β+2)/4}

and

\sup_{z\in\mathbb{K}}\mathbb{P}(|W_n^{\mathbb{K}}-z|\leq\varepsilon)\lesssim_βn^{(β+2)/4}\varepsilon^β

. In particular, for

\mathbb{K}=\mathbb{C}

, this resolves the Permanent Anticoncentration Conjecture of Aaronson and Arkhipov. The proof compares the squared Gaussian permanent with the squared (Study) determinant in Laplace-transform order.