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REVIEW 3 major objections 4 minor 55 references

Proof of Hiding Conjecture in Gaussian Boson Sampling

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves the hiding conjecture for Gaussian boson sampling in the maximal-squeezing regime, showing that an $N\times N$ corner of an $M\times M$ circular orthogonal ensemble matrix approaches a complex symmetric Gaussian in total…

desk verdict The core theorem is real and the proof is sound; the only real caveat is that the hardness transfer to the GGT ensemble remains entrywise, so the abstract slightly oversells the GBS hardness implication. read the letter →

arxiv 2508.00983 v2 pith:UVDDKW36 submitted 2025-08-01 quant-ph cs.CCmath-phmath.MP

Laura Shou , Sarah H. Miller , Victor Galitski This is my paper · ORCID
classification quant-phcs.CCmath-phmath.MP MSC 60B2081P68
keywords GaussianbosonsamplinghidingconjecturecircularorthogonalensembletotalvariationdistanceKullback-Leiblerdivergencerandommatrixsubmatrixhafnianquantumcomputationaladvantage
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves the hiding conjecture for Gaussian boson sampling when every input mode is squeezed ($K = M$), the regime realized in recent large-scale experiments. The result is quantitative: for an $N\times N$ corner $A$ of an $M\times M$ random symmetric unitary (COE) matrix, with $N = o(\sqrt{M})$, the rescaled matrix $\sqrt{M} A$ is within total variation distance $O(N/\sqrt{M})$ of a complex symmetric Gaussian matrix $G$. That closeness is exactly what allows one to hide a hard random hafnian instance inside the output distribution of a GBS device, so a sampler cannot identify and corrupt it. The authors also prove a multiplicative density bound needed for the instance-generating hardness argument, and extend the sparse-regime hiding theorem to its conjectured maximal submatrix size. The upshot is that the hardness argument for simulating GBS in the experimentally relevant all-squeezed regime now stands on proof rather than on the hiding conjecture.

What carries the argument

The load-bearing object is the explicit submatrix density of a COE matrix, quoted from the paper's reference [21]: $f(Z) \propto \det(I_N - Z^\dagger Z)^{(M-2N-1)/2}$ on the support $\lambda_{\max}(Z^\dagger Z) < 1$. After rescaling to $\sqrt{M}A$, this density becomes a product of factors $(1-\lambda_j/M)^{(M-2N-1)/2}$, which for small singular values approximates the Gaussian density $\exp(-\operatorname{Tr}(Z^\dagger Z)/2)$. The proof machinery is the Kullback-Leibler divergence method from [25]: bound the relative entropy between the COE-corner density and the Gaussian density using singular-value moment estimates obtained by Weingarten calculus, then convert to total variation distance via Pinsker's inequality. Because the normalizing constant of the COE density is left undetermined, a separate argument shows it is within $1+O(N^2/M)$ of one, using the fact that both densities integrate to one.

What would settle it

Compute the Kullback-Leibler divergence of Section 3 numerically for moderate $M$ with $N = c\sqrt{M}$ using the quoted density (Theorem 2.1) and the Gaussian density, and check the claimed $O(N^2/M)$ scaling; if the divergence grows faster than $N^2/M$ as $M$ increases, the bound in Theorem 1.2 would fail. Alternatively, a Monte Carlo estimate of $d_{\mathrm{TV}}(\sqrt{M}A,G)$ from sampled COE corners for $N = c\sqrt{M}$ should stay bounded by a constant times $c$.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: if $A$ is the upper-left $N\times N$ submatrix of an $M\times M$ circular orthogonal ensemble (COE) matrix and $G$ is an $N\times N$ complex symmetric Gaussian with $\mathcal{CN}(0,2)$ diagonal entries and $\mathcal{CN}(0,1)$ off-diagonal entries, then for $N = o(\sqrt{M})$ the total variation distance between $\sqrt{M} A$ and $G$ is at most $O(N/\sqrt{M})$. Because $K = M$ makes the hidden submatrix $U_{N,M}U_{N,M}^\mathsf{T}$ exactly a COE submatrix, this proves the hiding conjecture for the maximal number of squeezed states. The proof bounds the Kullback-Leibler divergence between the two densities and applies Pinsker's inequality; a notable technical feature is that the normalizing constant of the COE submatrix density is not evaluated, with a separate proposition showing that the unknown constant is close to one. The paper further establishes the multiplicative density estimate $f(Z) \le (1+O(N^3/M))g(Z)$ for $N = o(M^{1/3})$, and shows that the two natural Gaussian target distributions $G$ and $GG^\mathsf{T}$ are entrywise close for $N = o(\sqrt{K}/\log K)$.

Load-bearing premise

The entire proof leans on a previously derived formula for the distribution of a square corner of a random symmetric unitary matrix, and if that formula's exponent, support condition, or $N \le M/2$ constraint is incorrect, the total-variation bound collapses.

Editorial extensions

If this is right

  • The hiding conjecture for GBS with $K = M$ holds with the conjectured maximal size $N = o(\sqrt{M})$, so this step in the hardness argument is no longer conjectural.
  • The quantitative bound $d_{\mathrm{TV}}(\sqrt{M}A, G) \le O(N/\sqrt{M})$ gives the rate at which the hidden submatrix becomes Gaussian, matching the rate conjectured for maximal submatrix size.
  • The multiplicative density estimate $f \le (1+O(N^3/M))g$ for $N = o(M^{1/3})$ supplies the instance-generating condition needed for the conventional boson-sampling-style hardness argument.
  • Corollary 1.1 shows that for $N = o(\sqrt{K}/\log K)$ the distributions $G$ and $GG^\mathsf{T}$ are entrywise close, so hardness of approximating $\operatorname{Haf}(G)$ transfers to $\operatorname{Haf}(GG^\mathsf{T})$ in that regime.
  • In the sparse regime $NK = o(M)$, the paper extends hiding to $d_{\mathrm{TV}}(MU_{NK}U_{NK}^\mathsf{T}, G_{NK}G_{NK}^\mathsf{T}) = O(\sqrt{NK/M})$, which also upgrades Fock boson sampling hiding to the maximal $N = o(\sqrt{M})$ size with a quantitative rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not prove $K < M$, but its $K = M$ result suggests the same total-variation statement likely holds for intermediate squeezing fractions, since the $K = M$ case is the one where the submatrix rows are most constrained.
  • The technique of skipping the normalizing constant may transfer to other random matrix submatrix densities, such as Haar orthogonal or unitary submatrices, where explicit normalizing constants are often unavailable; this could simplify total-variation convergence proofs for other ensembles.
  • By proving hiding against the simple symmetric Gaussian $G$ rather than the correlated outer product $GG^\mathsf{T}$, and then showing the two targets are close, the paper suggests the average-case hardness question for GBS may be most naturally posed over symmetric Gaussian matrices.
  • A direct numerical check of the claimed $O(N^2/M)$ KL-divergence scaling for moderate $M$ with $N = c\sqrt{M}$ would be a cheap way to test whether the theorem's rate is tight in practice.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proves the hiding conjecture for Gaussian boson sampling (GBS) in the case of a maximal number of squeezed states, K=M. The main result, Theorem 1.2, states that for an N×N submatrix A of an M×M circular orthogonal ensemble (COE) matrix, with N=o(√M), the rescaled matrix √M A is within total variation distance O(N/√M) of a complex symmetric Gaussian matrix G∼Gsym_N. The proof adapts the Jiang–Ma KL-divergence method to the COE submatrix density, avoiding explicit evaluation of the normalizing constant by introducing a carefully chosen constant ζ and bounding it in Proposition 3.2. The paper also proves Corollary 1.1, which shows that Gsym_N and GGT_NK are close in entrywise distance for N=o(√K/log K), and Theorem 1.3, a multiplicative density-ratio bound needed for the instance-generating method. Appendix A extends the sparse hiding result for GBS to K=o(M).

Significance. If the proof is correct, this is the first rigorous total-variation hiding result for GBS in the experimentally relevant regime K∝M, and it places the hardness argument for GBS with a maximal number of squeezed states on a footing comparable to that of Aaronson–Arkhipov boson sampling. The paper is largely self-contained apart from the cited COE submatrix density, and the use of Weingarten calculus to compute singular value moments is a clear technical strength. However, the only internal cross-check of the load-bearing density, the Hua constant in Remark 1.1, is numerically wrong, and several presentation errors (notably Eq. (5.8)) need correction before the manuscript can be considered reliable.

major comments (3)
  1. [Remark 1.1, Eq. (1.7)] The stated normalization constant c′_{M,N} is incorrect at N=1. For N=1, Eq. (1.7) evaluates to (M−2)/(2π), but the density in (2.1) with exponent (M−3)/2 is normalized only by (M−1)/(2π). This is independently confirmed by the exact Weingarten computation in Eq. (4.9), which gives E|A_{11}|² = 2/(M+1) for the density c(1−|z|²)^{(M−3)/2}. Since Theorem 2.1 is the sole external input on which the entire KL-divergence argument rests, the authors must correct (1.7) or remove the claim, and ideally provide a derivation or a moment-based consistency check of (2.1) for general N.
  2. [Eq. (5.8)] The displayed expression does not follow from Eq. (5.7). The factor 1/2 multiplying the sum in the exponent of (5.7) is missing in the terms (1−(2N+1)/M)^{N(M−2N−1)} e^{2N²+N}. The final conclusion f/g ≤ 1+O(N³/M) is unaffected because the correct exponent is still O(N³/M), but the equality as written is wrong and should be fixed.
  3. [Section 5.1, proof of Corollary 1.1] The construction is confusingly written because G denotes both the N×K i.i.d. complex Gaussian matrix and the N×N Gsym matrix. In addition, the conditional distribution ρ_{G|X} used to couple √K W_NN with G is not explicitly defined; since ρ is an optimal coupling on a Polish space, the conditional distribution is well-defined, but the paper should spell this out. Finally, the condition N=o(√K/log K) should be derived from Eq. (5.5) by making explicit how the exponential term CK² e^{−cε√K/N} forces the logarithmic factor.
minor comments (4)
  1. [Introduction] There are typographical artifacts such as 'Section 1 1.1' and 'Section 1 1.3'; these should read 'Section 1.1' and 'Section 1.3'.
  2. [Lemma 4.2] The concentration inequality (4.6) for the operator norm of a symmetric Gaussian matrix should be checked against the cited reference [41, Theorem 2.26] for the correct Lipschitz constant; the text says the norm is √2-Lipschitz, but the usual result for symmetric matrices is 1-Lipschitz with respect to the Frobenius norm.
  3. [Appendix A, Eq. (A.8)-(A.10)] The algebra leading to the O(pq/M) bound is dense; adding a short explanation of the cancellation of the pq²/2 terms would improve readability.
  4. [Footnote 2] The caveat that total variation convergence between G and GGT is not established is important and should be given more prominence in the main text, since the original GBS hiding conjecture [6] is often phrased in terms of GGT.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is self-contained given established external results, with no fitted inputs and no load-bearing self-citations.

full rationale

I walked the derivation chain of Theorem 1.2. The TV bound is obtained from Pinsker's inequality and KL-divergence estimates in Propositions 3.1 and 3.2. Proposition 3.1 starts from the COE submatrix density quoted as Theorem 2.1 from Friedman and Mello [21], and from singular-value moment estimates (Lemma 2.2) that the paper proves directly via Weingarten calculus; no parameter is fitted to the target Gaussian distribution. Proposition 3.2 handles the unknown normalizing constant by exploiting the fact that both f and g are probability densities, together with tail estimates for the singular values of G and of the COE submatrix. That is a normalization argument, not an input borrowed from the hiding conjecture. The paper deliberately does not compute the normalizing constant, so there is no 'fitted input called prediction' pattern. The switch from the GGT form of the conjecture to Gsym is explicit and is justified by Corollary 1.1, whose proof uses an independent embedding result of Jiang [23] together with Theorem 1.2; although Corollary 1.1 is derived from Theorem 1.2, this is a legitimate logical consequence, not a circular premise, since Theorem 1.2 is proved first without invoking GGT. The sparse-case extension in Appendix A likewise follows an external submatrix-density formula from [31], again without using the target result as input. I found no self-citations that carry the argument and no uniqueness claim imported from the authors' own prior work. The only potentially fragile point I noticed is a numerical consistency concern about the Hua constant quoted in Remark 1.1 at N=1, but even if that printed constant needs correction, it would be a correctness or calibration issue in a cross-check, not a circular derivation. Accordingly, no circular steps are present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem rests on established random matrix theory results: the COE submatrix density, Weingarten calculus asymptotics, and Jiang's entrywise approximation. The only unproved input for the hardness narrative is the average-case hardness of the hafnian for the symmetric Gaussian ensemble, which the paper explicitly marks as a conjecture.

assumptions (5)
  • standard math COE submatrix density formula (Friedman-Mello, Theorem 2.1)
    The proof uses f(Z) proportional to det(I_N - Z-dagger Z)^{(M-2N-1)/2} on the support lambda_max(Z-dagger Z) < 1 for the top-left N by N submatrix of an M by M COE matrix. This is the starting point of the KL-divergence computation in Section 3 and is not re-derived.
  • standard math Weingarten calculus asymptotic expansion (Collins-Sniady, Theorem 4.4)
    Lemma 2.2 derives the singular value moment estimates from the leading-order asymptotics of the Weingarten function; if the stated asymptotics failed, the KL bound in Proposition 3.1 would not hold.
  • standard math Jiang's entrywise approximation theorem for COE (Theorem 1 of [23])
    Corollary 1.1 uses the bound P[max |sqrt(K) W_ij - (ZZ^T)_ij| > t] ≤ C K^2 e^{-c t sqrt(K)/N} to transfer Theorem 1.2 to an entrywise comparison between Gsym and GGT.
  • domain assumption Average-case hardness of approximating Haf(G) for G ~ Gsym_N
    The claimed hardness application replaces the original GGT average-case hardness conjecture with a conjectured hardness for the symmetric Gaussian ensemble. This is explicitly stated as a conjecture and is not proved in the paper.
  • standard math Total variation and coupling theorems (Pinsker inequality, optimal coupling)
    These standard results convert KL divergence bounds to TV distance and allow the construction of joint distributions in Corollary 1.1.

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Pith. "Pith review of Proof of Hiding Conjecture in Gaussian Boson Sampling." pith.science (2026). https://pith.science/paper/UVDDKW36

@misc{pith2026250800983,
  author       = {Pith},
  title        = {Pith review of: Proof of Hiding Conjecture in Gaussian Boson Sampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UVDDKW36}},
  note         = {Machine review of arXiv:2508.00983}
}
abstract

Gaussian boson sampling (GBS) is a promising protocol for demonstrating quantum computational advantage. One of the key steps for proving classical hardness of GBS is the so-called ``hiding conjecture'', which asserts that one can ``hide'' a complex Gaussian matrix as a submatrix of the outer product of Haar unitary submatrices in total variation distance. In this paper, we prove the hiding conjecture for input states with the maximal number of squeezed states, which is a setup that has recently been realized experimentally [Madsen et al., Nature 606, 75 (2022)]. In this setting, the hiding conjecture states that a $o(\sqrt{M})\times o(\sqrt{M})$ submatrix of an $M\times M$ circular orthogonal ensemble (COE) random matrix can be well-approximated by a complex Gaussian matrix in total variation distance as $M\to\infty$. This is the first rigorous proof of the hiding property for GBS in the experimentally relevant regime, and puts the argument for hardness of classically simulating GBS with a maximal number of squeezed states on a comparable level to that of the conventional boson sampling of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013)].

Figures

Figures reproduced from arXiv: 2508.00983 by the authors.

Figure 1
Figure 1. FIG. 1. A schematic of the Gaussian boson sampling (GBS) setup. An initial Gaussian state with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of the submatrix sizes considered for Fock boson sampling (gray) and Gaussian boson [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Indices [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Analogue of Fig [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

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    INTRODUCTION AND MAIN RESUL TS The promise of quantum advantage is a key motivation behind quantum computing. However, prac- tically demonstrating quantum advantage with physical quantum computers remains a technological challenge. In this regard, Gaussian boson sampling (GBS), based on the boson sampling protocol of [1], has emerged as a promising method...

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