arXiv:2609.06526

Shifted Anticoncentration for Real Gram Hafnians and Symmetric Gaussian Hafnians

Hongru Zhao

math.PRquant-phSubmitted Sep 6, 2026• Announced Sep 8, 2026

Abstract

We prove a uniform shifted anticoncentration theorem for the hafnian of a real Gaussian Gram matrix under an explicit condition on the row dimension. After normalization by its root mean square, the law has a bounded continuous density, maximal at zero, and every interval has probability bounded by an explicit coefficient times its radius. Under suitable growth conditions on the row dimension, this coefficient grows at most polynomially in the hafnian order, meaning half the dimension of the Gram matrix. We also compute the exact second moment. The proof exploits the perfect matching structure, combining conditional Gaussian representations, row suspension, and bilinear interpolation to control an inverse moment of the conditional variance. At fixed hafnian order, a rescaled limit as the row dimension grows yields corresponding bounds for the hafnian of a real symmetric Gaussian matrix with independent entries above the diagonal.

Comments: 25 pages, 1 figure, 1 table

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arXiv:2609.06526

Shifted Anticoncentration for Real Gram Hafnians and Symmetric Gaussian Hafnians

Hongru Zhao

math.PRquant-phSubmitted Sep 6, 2026• Announced Sep 8, 2026

arXiv HTML PDF

Abstract

Comments: 25 pages, 1 figure, 1 table

Related papers

Local Anticoncentration for Gaussian Boson Sampling via Conditional Wishart Geometry

Hongru Zhao

Local anticoncentration controls the conversion of additive output probability estimates to relative estimates in Gaussian boson sampling. We prove a center-uniform quadratic small-ball bound for hafnians of complex Gaussian transpose Gram matrices for $2n$ 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.Local anticoncentration controls the conversion of additive output probability estimates to relative estimates in Gaussian boson sampling. We prove a center-uniform quadratic small-ball bound for hafnians of complex Gaussian transpose Gram matrices for $2n$ 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.

math.PRCross-listedarXiv:2610.00112Sep 10, 20260 citations

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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.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.

arXiv:2609.35019Sep 28, 20260 citations

Anticoncentration of the Permanent in Ginibre Ensembles

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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.

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Hongru Zhao

Gaussian boson sampling requires control of how closely finite optical matrices follow Gaussian reference laws. We prove an explicit total variation bound of order $N^2/M$ between a rescaled Haar transpose Gram block and its Gaussian transpose Gram counterpart, where $N$ is the detected photon count and $M$ is the number of optical modes. The bound establishes quantitative product hiding uniformly over every number of squeezed inputs. The proof combines Stiefel recursion, centered circular orthogonal ensemble scores, and a rectangular entropy estimate. Applications combine hiding with local hafnian bounds to obtain relative probability guarantees and connect them to sampler error through exact photon-sector normalization.Gaussian boson sampling requires control of how closely finite optical matrices follow Gaussian reference laws. We prove an explicit total variation bound of order $N^2/M$ between a rescaled Haar transpose Gram block and its Gaussian transpose Gram counterpart, where $N$ is the detected photon count and $M$ is the number of optical modes. The bound establishes quantitative product hiding uniformly over every number of squeezed inputs. The proof combines Stiefel recursion, centered circular orthogonal ensemble scores, and a rectangular entropy estimate. Applications combine hiding with local hafnian bounds to obtain relative probability guarantees and connect them to sampler error through exact photon-sector normalization.

quant-pharXiv:2609.01008Sep 2, 20260 citations

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.

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.

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.

2k

-th moment of

N\times N

matrices with independent entries, the dominant saddle point is replica symmetric for

k>2

, and we obtain in closed form the exponential growth rate, the prefactor and the first

1/N

correction, in agreement with exact enumerations. The growth rate depends only on whether the entries are real or complex; higher cumulants affect only the prefactor and the

1/N

term. At

k=2

the replica-symmetric solution becomes unstable, separating a power-law bulk of the distribution from a large-deviation tail, which we compute. The results apply to boson sampling with single photons and with squeezed light.

Gaussian boson sampling requires control of how closely finite optical matrices follow Gaussian reference laws. We prove an explicit total variation bound of order

N^2/M

between a rescaled Haar transpose Gram block and its Gaussian transpose Gram counterpart, where

N

is the detected photon count and

M

is the number of optical modes. The bound establishes quantitative product hiding uniformly over every number of squeezed inputs. The proof combines Stiefel recursion, centered circular orthogonal ensemble scores, and a rectangular entropy estimate. Applications combine hiding with local hafnian bounds to obtain relative probability guarantees and connect them to sampler error through exact photon-sector normalization.