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arXiv:2610.00112

Local Anticoncentration for Gaussian Boson Sampling via Conditional Wishart Geometry

Hongru Zhao

math.PRquant-phSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
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Abstract

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.

Comments
39 pages, 4 figures, 1 table. Includes all appendices

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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.

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Exact Moments of Gaussian Gram Hafnians Reveal an $n^2/\log n$ Threshold for Weak Anticoncentration

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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.

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Anticoncentration and entanglement in Gaussian boson sampling

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Anticoncentration and entanglement are two properties used to study classical hardness or easiness of Gaussian boson sampling in varying regimes. In this paper, we consider three directions concerning anticoncentration and entanglement in Gaussian boson sampling: (1) We derive closed-form expressions for the second moment of hafnians of symmetric Gaussian products, and use this to precisely locate the anticoncentration transition as a function of the number of squeezed input modes. (2) We derive closed-form expressions for the Rényi-$α$ 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.

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Proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezed input modes

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Gaussian boson sampling (GBS) is a sampling task proposed to demonstrate quantum advantage. We consider Gaussian boson sampling on $M$ optical modes, with $K$ equally squeezed input modes and $N$ observed photon counts. We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers $K$, which is a part of the argument for classical hardness of GBS. In particular, we show that for any $K$ and $N=o(\sqrt{K})$, the symmetric product $MK^{-1/2}U_{NK}U_{NK}^T$, for $U_{NK}$ the top left $N\times K$ submatrix of an $M\times M$ Haar random unitary $U$, is close in total variation distance to both an $N\times N$ symmetric complex Gaussian matrix $\mathbf G$ with independent entries, and the symmetric product $GG^T/\sqrt{K}$ for $G$ an $N\times K$ matrix of iid standard complex Gaussians. We show however that the density-based instance generating method of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8] used to efficiently implement a hiding procedure fails for Gaussian boson sampling with $K=cM$ if $c<1/2$. Instead we use approximate instance generating to implement the hiding for the usual classical hardness reduction.

quant-pharXiv:2608.19314•Aug 21, 2026•0 citations
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 2n2n2n selected modes, with an explicit finite coefficient that is polynomial when the number of Gaussian rows is at least of order n2/log⁡nn^2/\log nn2/logn. 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.
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.
Gaussian boson sampling requires control of how closely finite optical matrices follow Gaussian reference laws. We prove an explicit total variation bound of order N2/MN^2/MN2/M between a rescaled Haar transpose Gram block and its Gaussian transpose Gram counterpart, where NNN is the detected photon count and MMM 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.
Let G2nG_{2n}G2n​ be a complex symmetric random matrix whose entries above the diagonal are independent standard circular complex Gaussians, and let Hn=haf⁡(G2n)H_n=\operatorname{haf}(G_{2n})Hn​=haf(G2n​). We prove the uniform shifted anticoncentration bound Pr⁡ ⁣(∣Hn(2n−1)!!−z∣≤ε)≤2nπ ε2 \Pr\!\left( \left| \frac{H_n}{\sqrt{(2n-1)!!}}-z \right| \le \varepsilon \right) \le 2\sqrt{\frac nπ}\,\varepsilon^2 Pr(​(2n−1)!!​Hn​​−z​≤ε)≤2πn​​ε2 for every z∈Cz\in\mathbb Cz∈C and ε>0\varepsilon>0ε>0. This establishes a local anticoncentration property that supports hardness arguments for quantum advantage in Gaussian boson sampling.
Exact Moments of Gaussian Gram Hafnians Reveal an n2/log⁡nn^2/\log nn2/logn Threshold for Weak Anticoncentration
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 Hk,n=haf(XTX)H_{k,n}=\mathrm{haf}(X^{\mathsf T}X)Hk,n​=haf(XTX), where X∈Ck×2nX\in\mathbb{C}^{k\times 2n}X∈Ck×2n has independent standard circular complex Gaussian entries. We evaluate E∣Hk,n∣2\mathbb{E}|H_{k,n}|^2E∣Hk,n​∣2 and E∣Hk,n∣4\mathbb{E}|H_{k,n}|^4E∣Hk,n​∣4 exactly by reducing four hafnian copies to a rank two Gaussian integral. For Rk,n=(E∣Hk,n∣2)2/E∣Hk,n∣4R_{k,n}=(\mathbb{E}|H_{k,n}|^2)^2/\mathbb{E}|H_{k,n}|^4Rk,n​=(E∣Hk,n​∣2)2/E∣Hk,n​∣4, we obtain Rk,n=4−n(2nn)/Fk,nR_{k,n}=4^{-n}\binom{2n}{n}/F_{k,n}Rk,n​=4−n(n2n​)/Fk,n​, where Fk,n=3F2(−n,−n,1/2;1,k/2;1)F_{k,n}={}3F_2(-n,-n,1/2;1,k/2;1)Fk,n​=3F2​(−n,−n,1/2;1,k/2;1) is a terminating generalized hypergeometric polynomial. If k/n2→c>0k/n^2\to c>0k/n2→c>0, then Fk,n→e1/cI0(1/c)F{k,n}\to e^{1/c}I_0(1/c)Fk,n→e1/cI0​(1/c), where I0I_0I0​ is the modified Bessel function of the first kind of order zero, and consequently Rk,nπn→[e1/cI0(1/c)]−1R_{k,n}\sqrt{πn}\to[e^{1/c}I_0(1/c)]^{-1}Rk,n​πn​→[e1/cI0​(1/c)]−1. Thus k≍n2k\asymp n^2k≍n2 is a smooth Bessel crossover, whereas the scaling order boundary for inverse polynomial weak anticoncentration is k≍n2/log⁡nk\asymp n^2/\log nk≍n2/logn. These conclusions concern the Gaussian surrogate moment criterion; finite dimensional Haar moment transfer and high probability small ball anticoncentration remain separate problems.
Anticoncentration and entanglement are two properties used to study classical hardness or easiness of Gaussian boson sampling in varying regimes. In this paper, we consider three directions concerning anticoncentration and entanglement in Gaussian boson sampling: (1) We derive closed-form expressions for the second moment of hafnians of symmetric Gaussian products, and use this to precisely locate the anticoncentration transition as a function of the number of squeezed input modes. (2) We derive closed-form expressions for the Rényi-ααα Page curves for Gaussian boson sampling. (3) We study unequal input squeezing parameters (si)i(s_i)_i(si​)i​, and prove estimates as well as monotonicity of the average-case Rényi-2 entropy Page curve in terms of the magnitudes ∣si∣|s_i|∣si​∣, which allow for extending equal squeezing results to unequal squeezing.
Gaussian boson sampling (GBS) is a sampling task proposed to demonstrate quantum advantage. We consider Gaussian boson sampling on MMM optical modes, with KKK equally squeezed input modes and NNN observed photon counts. We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers KKK, which is a part of the argument for classical hardness of GBS. In particular, we show that for any KKK and N=o(K)N=o(\sqrt{K})N=o(K​), the symmetric product MK−1/2UNKUNKTMK^{-1/2}U_{NK}U_{NK}^TMK−1/2UNK​UNKT​, for UNKU_{NK}UNK​ the top left N×KN\times KN×K submatrix of an M×MM\times MM×M Haar random unitary UUU, is close in total variation distance to both an N×NN\times NN×N symmetric complex Gaussian matrix G\mathbf GG with independent entries, and the symmetric product GGT/KGG^T/\sqrt{K}GGT/K​ for GGG an N×KN\times KN×K matrix of iid standard complex Gaussians. We show however that the density-based instance generating method of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8] used to efficiently implement a hiding procedure fails for Gaussian boson sampling with K=cMK=cMK=cM if c<1/2c<1/2c<1/2. Instead we use approximate instance generating to implement the hiding for the usual classical hardness reduction.