{"id":"bd2b3e73-bdbf-4563-bf80-585bcce24ac7","arxiv_id":"2508.00983","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The top-left N by N submatrix of an M by M circular orthogonal ensemble random matrix, scaled by sqrt(M), converges to a complex symmetric Gaussian matrix in total variation distance for N much smaller than sqrt(M).","lead":"This paper proves a long-standing 'hiding' conjecture in Gaussian boson sampling: for the experimentally relevant case where every input mode is squeezed, a small submatrix of the random interference matrix is statistically indistinguishable from a complex Gaussian matrix. This was the missing mathematical step for a rigorous argument that classical computers cannot efficiently simulate the current generation of Gaussian boson sampling experiments.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.2 rests entirely on the cited COE submatrix density (2.1) with exponent (M−2N−1)/2; the paper's only cross-check, Hua's constant (1.7), fails the N=1 test ((M−2)/(2π) vs required (M−1)/(2π)), so the load-bearing N-coefficient is not independently confirmed.","rationale":"I read the paper in good faith and worked through the main computational chain. The reduction at K=M is correct: √M·(top-left N×N of UU^T) is the right object, and the TV comparison to G_sym is the right statement. Proposition 3.1's bound E log(˜f/g) ≤ O(N²/M) is correct, but it depends on an exact N³/M cancellation: the +N³/M from (2N+1)(N²+N)/(2M) cancels the two −N³/(2M) contributions (from ζ and from the second-moment term), leaving O(N²/M) + O(N⁴/M²); the paper's display '= O(N²/M)' skips this. Proposition 3.2 checks out: signs in (3.13), the tail bounds (3.14)–(3.15), the lower bound (3.17) on log(1−x) for x ≤ 1/2, and the final ζ bounds all close for N = o(√M). I also resolved the apparent scale problems: with the stated convention CN(0,σ²) (σ² the variance), the GGT_NK entries have variance K·(1/√K)² = 1, matching G_sym, so Corollary 1.1 and Theorem 1.5 are internally consistent; the factor error in (5.8) doubles both the log-exponent and the exponential factor and does not affect the O(N³/M) conclusion of Theorem 1.3; the coupling construction in Corollary 1.1 is valid though underexplained. The one input I could not independently confirm is Theorem 2.1 itself, and the paper's own cross-check of it (Eq. (1.7)) has an N=1 discrepancy: (1.7) gives c = (M−2)/(2π), whereas the exact density of the (1,1) COE entry, pinned by E|(UU^T)_11|² = 2/(M+1), requires c = (M−1)/(2π). Because the N-coefficient in the exponent is load-bearing (an O(N)-error would inject a non-cancelling O(N³/M) into the KL and break the TV rate), this is the single most load-bearing concern, matching the reader's weakest_assumption and sharpening it with a concrete internal inconsistency. The proposed N=2 analytic/numerical check settles it. I also note the paper's own footnote 2 concedes that even TV convergence of G and GGT would not imply convergence of hafnian moments, so the advertised hardness application ('comparable level' to Aaronson–Arkhipov) relies on a new average-case assumption for Haf(G_sym); this is a limitation of the framing, not of Theorem 1.2. Verdict: UNCHANGED — CONDITIONAL acceptance remains the right call, pending independent verification of (2.1).","tokens_in":22147,"tokens_out":63753,"duration_ms":619357,"concrete_test":"Verify Theorem 2.1 at N=2 without relying on [21]. (i) Analytic: for a 2×2 complex symmetric Z, use the eigenvalue/unitary decomposition to compute M_2 := ∫ |Z_12|² · c · det(I_2 − Z†Z)^{(M−5)/2} 1_{λmax<1} dZ and check M_2 = 1/(M+1), the exact Weingarten value E|(UU^T)_12|²; separately check ∫ f(Z) dZ = 1 and compare the implied c with a corrected version of (1.7). (ii) Numerical: for M ∈ {32, 64, 128}, sample 10^4 Haar unitaries U, set H = UU^T, extract the top-left 2×2 submatrix √M A, and run a bin-based χ² goodness-of-fit against (2.1) for the exponents (M−5)/2, (M−7)/2, and (M−3N−1)/2. If only (M−5)/2 matches, the load-bearing N-coefficient is confirmed and the Theorem 1.2 rate stands; a match to (M−7)/2 would indicate the paper's M-coefficient needs rechecking and Remark 1.1's off-by-one is in (2.1), not (1.7).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim d_TV(√M A, G) ≤ O(N/√M) is obtained by bounding the KL divergence in Propositions 3.1 and 3.2, both of which start from the rescaled density (3.3), i.e., from Theorem 2.1's functional form f(Z) ∝ det(I_N − Z†Z)^{(M−2N−1)/2} on λmax(Z†Z) < 1. The proof is sensitive to the exact exponent: an O(1) shift of the constant (e.g., (M−2N−2)/2) only shifts E log(˜f/g) by O(N²/M) and is absorbable, but a change in the N-coefficient (e.g., (M−3N−1)/2) injects an extra ±N³/(2M)-type term into (3.9) that does not cancel against the chosen ζ = exp(−N³/(2M)); the KL would then scale as O(N³/M) and the TV bound would become O(N^{3/2}/M^{1/2}), which need not vanish for N = o(√M). The support indicator is likewise load-bearing for the set S = {λmax ≤ M/2} used in (3.13). Theorem 2.1 is cited from [21] and not re-derived, and the paper's stated cross-check (Remark 1.1, Eq. (1.7)) fails a direct consistency test: at N=1, (1.7) evaluates to (M−2)/(2π), whereas the exact Weingarten computation E|(UU^T)_11|² = 2/(M+1) requires the normalized density c(1−|z|²)^{(M−3)/2} to have c = (M−1)/(2π). So (1.7) as printed is off by (M−2)/(M−1), and the remark does not close the loop on (2.1). I verified the rest of the chain: the O(N²/M) bound in (3.9) does hold, but only through the exact cancellation of (2N+1)(N²+N)/(2M) against the two −N³/(2M) terms, a step the paper hides inside '= O(N²/M)'; Proposition 3.2's ζ bounds, the tail estimates, and the coupling in Corollary 1.1 are internally consistent. The unresolved input is therefore exactly (2.1), whose exponent must be independently checked.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":22691,"tokens_out":20851,"duration_ms":193510,"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":[{"comment":"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.","section":"Remark 1.1, Eq. (1.7)"},{"comment":"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.","section":"Eq. (5.8)"},{"comment":"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.","section":"Section 5.1, proof of Corollary 1.1"}],"minor_comments":[{"comment":"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'.","section":"Introduction"},{"comment":"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.","section":"Lemma 4.2"},{"comment":"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.","section":"Appendix A, Eq. (A.8)-(A.10)"},{"comment":"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.","section":"Footnote 2"}],"recommendation":"major_revision","confidential_remarks":"The wrong Hua constant in Eq. (1.7) is likely a transcription error, since the authors explicitly state they do not use it in the proof. However, it is the only internal check of the external density (2.1), and its failure means the load-bearing N-coefficient in the exponent is not independently confirmed by the paper. The authors should be asked to correct or remove (1.7) and to add a direct consistency check between (2.1) and the Weingarten-derived moments in Lemma 2.2. If that is done, the main theorem is likely sound, but the current version should not be accepted as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is a genuine advance: first TV-distance proof of the hiding conjecture for GBS with K=M, i.e., that an N×N submatrix of an M×M COE matrix, rescaled by sqrt(M), is within O(N/sqrt(M)) of a complex symmetric Gaussian in total variation, for N=o(sqrt(M)). That is exactly the strong sense needed for the hardness reduction, and it was open. The proof adapts Jiang–Ma cleanly, and the trick of skipping the normalizing constant via Proposition 3.2 is nice and likely reusable.\n\nI checked the stress-test note carefully, and its main concern does not land. The N=1 check actually confirms the exponent in Theorem 2.1: with exponent (M-3)/2, the density normalizes to (M-1)/(2π), and the resulting second moment 2/(M+1) matches the Weingarten calculation. The Hua constant quoted in Remark 1.1 is off by (M-2)/(M-1) at N=1, but that remark is not used in the proof; it is a misprint or a constant for a different normalization, not a load-bearing flaw. The KL cancellation in (3.9) works as written, and (5.8) is also fine: the log is (7/2)N^3/M + O(N^2/M), still O(N^3/M).\n\nWhere the paper is genuinely softer: it proves hiding for the symmetric Gaussian ensemble Gsym, while the original conjecture and the GBS hardness argument want GGT of an i.i.d. Gaussian. Corollary 1.1 only gives entrywise closeness, not TV closeness, between G and GGT for N=o(sqrt(K)/log K). The authors say they expect TV closeness but do not prove it. So the hardness argument now rests on a new average-case assumption for Haf(G) rather than Haf(GGT). That is worth stating clearly; the abstract's \"comparable level\" claim is a bit too strong. The proof also leans on the cited Friedman–Mello submatrix density without re-deriving it, which is standard, and the N=1 consistency check supports that citation.\n\nMinor: Corollary 1.1's construction overloads the letter G and is hard to follow, but it is fixable and the argument checks out. Some small exposition issues around Theorem 1.3's N=o(M^{1/3}) regime are worth tightening.\n\nThis paper deserves a serious referee. It resolves the central open step for the experimentally relevant K=M regime, the mathematical core is sound, and the remaining gap (Gsym vs GGT in TV) is clearly flagged. I would send it to peer review and would cite it. Bring it to reading group.","headline":"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.","tokens_in":23256,"tokens_out":5519,"would_cite":true,"duration_ms":60252,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","81P68"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Gaussian boson sampling","hiding conjecture","circular orthogonal ensemble","total variation distance","Kullback-Leibler divergence","random matrix submatrix","hafnian","quantum computational advantage"],"falsifier":"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 $c$.","tokens_in":21934,"feed_emoji":"🎲","tokens_out":9591,"duration_ms":88418,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hiding conjecture for Gaussian boson sampling proved at K=M","feed_subtitle":"A submatrix of a random symmetric unitary matrix is shown Gaussian in total variation, closing a hardness gap for GBS.","key_machinery":"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.","core_discovery":"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 $N = o(\\sqrt{K}/\\log K)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the explicit submatrix density of a COE matrix (Theorem 2.1) on which the entire KL-divergence computation rests.","marker":"[21]"},{"why":"Supplies the KL-divergence method and the structure of singular-value moment bounds used to prove Theorem 1.2.","marker":"[25]"},{"why":"Provides the hiding framework, the instance-generating method, and the technique of leaving the normalizing constant undetermined.","marker":"[1]"},{"why":"Provides the entrywise closeness between a COE submatrix and a Gaussian outer product used to prove Corollary 1.1.","marker":"[23]"},{"why":"Supplies the unitary submatrix density used in Appendix A to prove the sparse-regime hiding theorem and the maximal Fock hiding bound.","marker":"[31]"},{"why":"Provides the Weingarten calculus asymptotics used to compute the singular-value moment estimates in Lemma 2.2.","marker":"[38]"},{"why":"Provides Pinsker's inequality converting the KL-divergence bound into the total-variation bound.","marker":"[39]"},{"why":"Contains the evaluation of the normalizing constant that the proof deliberately avoids, noted as an alternative to Proposition 3.2.","marker":"[26]"}],"fun_headline_variants":["Hiding conjecture proven for maximal-squeezing GBS","GBS hardness gap closed: hiding conjecture proved","First rigorous proof of hiding property for GBS","Hiding conjecture settled for Gaussian boson sampling","Maximal squeezing: hiding conjecture now proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hiding conjecture proven for maximal-squeezing GBS","GBS hardness gap closed: hiding conjecture proved","First rigorous proof of hiding property for GBS","Hiding conjecture settled for Gaussian boson sampling","Maximal squeezing: hiding conjecture now proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000333,"raw_usage":{"total_tokens":1909,"prompt_tokens":1061,"completion_tokens":848,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":677,"completion_tokens_details":{"reasoning_tokens":776}},"tokens_in":677,"tokens_out":848,"duration_ms":8588,"temperature":1.0,"reasoning_tokens":776,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:58:59.062783+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $c$.","supporting_citations":[{"cited_title":"However, these weaker forms of convergence are not sufficient for the hiding property and subsequent hardness argument [1, 7]","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit submatrix density of a COE matrix (Theorem 2.1) on which the entire KL-divergence computation rests."},{"cited_title":"Ehrenberg, J","cited_arxiv_id":null,"evidence_quote":"Supplies the KL-divergence method and the structure of singular-value moment bounds used to prove Theorem 1.2."},{"cited_title":"Proof of Hiding Conjecture in Gaussian Boson Sampling","cited_arxiv_id":"2508.00983","evidence_quote":"Provides the hiding framework, the instance-generating method, and the technique of leaving the normalizing constant undetermined."},{"cited_title":"Grier, D","cited_arxiv_id":null,"evidence_quote":"Provides the entrywise closeness between a COE submatrix and a Gaussian outer product used to prove Corollary 1.1."},{"cited_title":"Jiang, How many entries of a typical orthogonal matrix can be approximated by independent normals?, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the unitary submatrix density used in Appendix A to prove the sparse-regime hiding theorem and the maximal Fock hiding bound."},{"cited_title":"Jiang, Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles, Probab","cited_arxiv_id":null,"evidence_quote":"Provides the Weingarten calculus asymptotics used to compute the singular-value moment estimates in Lemma 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Pinsker's inequality converting the KL-divergence bound into the total-variation bound."},{"cited_title":"Ehrenberg, J","cited_arxiv_id":null,"evidence_quote":"Contains the evaluation of the normalizing constant that the proof deliberately avoids, noted as an alternative to Proposition 3.2."}],"review_version":1}