{"id":"a6f3bc2f-8e07-423e-8c60-fe10f276c282","arxiv_id":"1908.01462","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper proposes quantum LS-SVM algorithms based on continuous-variable matrix inversion and sparse hybrid solutions, but the core equations contain a sign error and an invalid unitary factorization.","lead":"This paper proposes two quantum algorithms for least-square support vector machines: one simplifies matrix inversion with continuous-variable quantum states, and the other creates sparse solutions using a hybrid quantum-classical pipeline. A generalist might read it because it claims exponential speed-ups for machine learning, but the central equations contain a sign error and an invalid operator split.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (10) factorizes a noncommuting matrix exponential into a product, so the phase-estimation unitary for M_s is not implemented; no Trotter bound is supplied. This alone breaks the claimed exponential speed-up. The Eq. (7) sign issue in the reader report is not confirmed by direct integration.","rationale":"The central claim is that LS-SVM training and inference can be done with logarithmic dependence on M via a simplified CV-assisted matrix inversion. That claim requires an implementable, efficient unitary that maps |u_i>|q>|g> to |u_i> e^{iλ_i p1 p2}|q>|g>, i.e., e^{iM_s p1 p2}. Eq. (10) is the only proposed construction. Since G1 and G3 do not commute, the product formula is not exact unless a Trotter approximation is supplied and analyzed; the paper provides neither. A Trotter approximation would introduce additional error and likely change the complexity, but the paper's stated O(ε^{-1}log(MN)) rests on the false exact identity. This is not a matter of convention or consensus; it is an internal algebraic inconsistency. The QSLS-SVM algorithm inherits the same CV matrix toolbox and sparse-solution claims, so it falls with the same defect. The reader's second criticism of Eq. (7) appears to be misplaced: evaluating the Gaussian integral for a scalar λ gives a positive factor proportional to 1/λ, not a sign flip. The prefactor may be 2 rather than 1, but that does not reverse labels. I therefore partially agree with the reader: the noncommutation of Eq. (10) is the load-bearing failure, while the sign-error claim is not confirmed. I recommend keeping the reader's REJECT verdict, because even setting aside Eq. (7), the unsupported factorization in Eq. (10) is sufficient to invalidate the proposed quantum speed-up.","tokens_in":13910,"tokens_out":10917,"duration_ms":109663,"concrete_test":"For a fixed 3x3 example with K=[[2,1],[1,1]] and γ=1, form M_s=G1+G2+G3. Compute D(t)=||e^{iM_s t}-e^{iG1 t}e^{iG2 t}e^{iG3 t}||_2 for t=1 and for t tending to 0. If D(t) is not identically zero and scales as O(t^2), Eq. (10) is false as an operator identity; then estimate the Trotter step count needed to achieve error ε and compare that cost with O(ε^{-1}log(MN)). This single computation decides whether the claimed phase-estimation unitary can be implemented at the stated cost.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is the asserted factorization in Eq. (10). With G1=blkdiag(K,0), G2=blkdiag(γ^{-1}I,0), and G3=[[0,1^T],[1,0]], the sum G1+G2+G3 equals M_s, but G1 and G3 do not commute, and generically G2 and G3 do not commute either. Therefore e^{iM_s p1 p2} ≠ ∏_{k=1}^3 e^{iG_k p1 p2} as an operator identity; the equality used for the HVQ-SVM phase-estimation step is false for any nonconstant kernel matrix K. No Trotter-Suzuki decomposition, error bound, or norm estimate is given, so the claimed cost O(ε^{-1}log(MN)) for constructing the inversion unitary has no basis. If the product is implemented instead, the eigenvalues seen by the CV modes are not those of M_s, and the post-selected state is not M_s^{-1}|y>. This invalidates the central subroutine and the exponential speed-up of both HVQ-SVM and QSLS-SVM, which inherits the same CV framework. A second point: direct evaluation of the scalar integral in Eq. (7) for an eigenvalue λ gives 2/λ, preserving the sign of λ; the reader's sign-inversion claim is therefore not supported. The normalization prefactor is off by 2, but a global positive factor does not reverse classification labels. The noncommutation issue in Eq. (10) is sufficient for rejection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes two quantum algorithms for least-squares support vector machines. The first (HVQ-SVM) aims to simplify the quantum LS-SVM of Rebentrost et al. by replacing the qubit-based matrix-inversion subroutine with a continuous-variable integral representation of M_s^{-1}, claiming evolution cost O(ε^{-1} log(MN)). The second (QSLS-SVM) is a hybrid scheme that uses QPCA/QSVT to reduce the dataset to R principal components and then solves an R-dimensional classical LS-SVM, claiming complexity O(N + R^3 + R ε^{-1} log(MN)). The paper also discusses polynomial and Gaussian kernel state preparation and interprets finite-width effects as regularization.","tokens_in":14251,"tokens_out":9863,"duration_ms":99188,"significance":"If the central subroutine were valid, the logarithmic dependence on the number of samples would be a meaningful advance for quantum SVM, and the hybrid sparse extension is a reasonable direction. The paper correctly identifies the dense, non-sparse structure of M_s as the obstacle to direct Hamiltonian simulation and proposes a block decomposition. It also provides an appendix connecting finite-squeezing errors to L2-type regularization. However, the validity of the main algorithm rests on the exact factorization of a matrix exponential of noncommuting matrices, which is false, and the post-selection step is not justified; therefore the claimed speedups are not established.","major_comments":[{"comment":"The identity e^{i M_s p1 p2} = ∏_{k=1}^3 e^{i G_k p1 p2} is false. Since M_s = G1 + G2 + G3, the equality would require the three Hermitian matrices to commute pairwise; for a generic kernel matrix K, [G1,G3] ≠ 0 and [G2,G3] ≠ 0. The text provides no Trotter-Suzuki decomposition, no ordering prescription, and no error bound. Because step 2 of the algorithm performs phase estimation with this unitary, the eigenphases read out in Eq. (11) are not those of M_s, and the post-selected state is not proportional to M_s^{-1}|ψ_y>. This invalidates the central HVQ-SVM subroutine and the claimed O(ε^{-1} log(MN)) complexity.","section":"Sec. III.A, Eq. (10)"},{"comment":"The ideal CV post-selection is not justified. Direct evaluation of the scalar integral in Eq. (7) for an eigenvalue λ > 0 gives +2π/λ, so the sign of λ is preserved; the advertised output λ_k^{-1} in Eq. (12) nevertheless lacks the corresponding normalization factor. More seriously, with ψ(p1) vanishing at p1 = 0 and g(p2) containing a factor p2, the amplitude for projecting onto |0_{p1}>|0_{p2}> is zero; the statement ⟨q|0_{p1}⟩ = ⟨r|0_{p2}⟩ = 1 is incorrect. The finite-width construction introduced in Appendix B is a different procedure, and its success probability, fidelity, and effect on the complexity are not incorporated into the main algorithm.","section":"Sec. III.A, Eqs. (7)-(12)"},{"comment":"The reduction to the top-R components is not established at the level needed for the claimed complexity. Taking Ω = U, one obtains K_Ω = U^T K U = diag(σ_1^2, ..., σ_M^2), so the top-R submatrix is diag(σ_i^2), not Σ_{R×R} as written in Eq. (14). Moreover, the error estimate in Eq. (35) bounds the discarded contribution to g(x̂) by a factor σ_{T+1}, but it does not bound the error in α_{ΩR} and b or the number of repetitions of the quantum measurement needed to achieve a target classification accuracy; hence the cost O(N + R^3 + R ε^{-1} log(MN)) is not derived.","section":"Sec. III.B, Eqs. (14) and (35)"}],"minor_comments":[{"comment":"The title contains an extraneous space in 'si mplified'; correct the typography.","section":"Title"},{"comment":"The vector y is first defined as y = (y1, ..., yM) with y ∈ R^{1×N}, but the labels y_i are M scalars; the dimension should be M, not N.","section":"Sec. II, Notations"},{"comment":"The sentence 'There are f(x_i)y_i = 1 - ξ_i' is grammatically unclear; it should state that the constraint in Eq. (3) implies f(x_i)y_i = 1 - ξ_i.","section":"Sec. II, Eq. (3)"},{"comment":"The expression for N_{x'}^{-1} is difficult to parse and appears to have inconsistent powers; please rewrite the normalization of |φ(x')> explicitly.","section":"Appendix A, Eq. (20)"},{"comment":"The phrase 'matrix Ms is full rank and unitary operation e^{iM_s p1 p2} can not be constructed' is confusing, since e^{iM_s t} is unitary for any Hermitian M_s; the authors likely mean that direct Hamiltonian simulation of the dense matrix M_s is unavailable.","section":"Sec. III.A, before Eq. (10)"},{"comment":"The cross-reference to 'Ref.[37]' does not match the citation numbering in the bibliography; check all cross-references.","section":"Appendix A, after Eq. (22)"}],"recommendation":"reject","confidential_remarks":"The noncommutation error in Eq. (10) is load-bearing and cannot be repaired by a local edit: the entire phase-estimation construction, the output state, and the complexity claim depend on the false factorization. A resubmission would need a complete Trotterized construction with explicit error bounds and a reworked complexity analysis, as well as a proper account of the CV post-selection step."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe useful thing to know: this is not the simplified quantum LS-SVM it claims to be. The central subroutine in HVQ-SVM, Eq. (10), asserts e^{iM_s p1 p2} = ∏_{k=1}^3 e^{iG_k p1 p2} with G_k = blkdiag(K,0), blkdiag(γ^{-1}I,0), and [[0,1^T],[1,0]]. Those matrices do not commute, so the operator identity is false. No Trotterization, no error bound. Since that exponential is the engine of the CV matrix-inversion step, both HVQ-SVM and QSLS-SVM inherit the gap. The claimed O(ε^{-1} log(MN)) cost rests on an unimplemented unitary.\n\nWhere the paper does something real: the assembled algorithms are new. CV-assisted matrix inversion applied to LS-SVM, and the hybrid sparse procedure using QPCA/QSVT, are not in the cited literature. The kernel feature-map construction that uses both qubits and CV modes is a genuinely different angle from Rebentrost et al. The appendix on finite-width effects as a form of L2 regularization is thoughtful, and the paper is honest about the need for thresholding in the sparse case.\n\nThe soft spots, in proportion: Eq. (10) is load-bearing and broken. The reader's report also flags a sign error in Eq. (7); my own quick integration did not confirm that — I get a different prefactor, not a clean sign flip that would reverse labels. So I would not hang the rejection on Eq. (7). The absence of numerical experiments is a real limitation for a hybrid proposal, though not the main reason for rejection. The complexity claims also ignore the dequantized baselines the paper itself cites and the readout overhead; the R dependence in O(N + R^3 + R ε^{-1} log(MN)) could swallow the speed-up when R is not tiny.\n\nBottom line: not publishable as is. The factorization error is mathematically clear and central. It could in principle be repaired with a Trotter-Suzuki decomposition and a proper error analysis, but that work is not here. A serious referee should still see it — partly because the error is instructive, partly because the CV approach might be salvageable. Worth engaging with if you work on CV quantum algorithms; otherwise you can let this one go.","headline":"The paper assembles new QML variants, but its central CV matrix-inversion step hinges on a false exponential factorization, so the speed-up claim lacks support.","tokens_in":14800,"tokens_out":7018,"would_cite":false,"duration_ms":68091,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Ac"],"model":"deepseek-v4-flash","headline":"The paper proposes quantum LS-SVM algorithms whose training cost scales as $O(\\varepsilon^{-1}\\log(MN))$ and whose hybrid sparse variant solves a reduced $R\\times R$ system after quantum singular-value extraction.","keywords":["quantum support vector machine","least-square SVM","continuous-variable quantum computing","quantum matrix inversion","sparse solution","quantum feature map","quantum principal component analysis","quantum speed-up"],"falsifier":"Evaluate Eq. (7) for a single positive eigenvalue $\\lambda$: the stated integral gives $-1/\\lambda$ rather than $\\lambda^{-1}$, which would make the inversion routine return the wrong sign. Separately, evaluate $e^{iM_s\\theta}$ against $e^{iG_1\\theta}e^{iG_2\\theta}e^{iG_3\\theta}$ for a small noncommuting kernel matrix; the difference is nonzero, and its size is the missing approximation cost in the claimed complexity.","tokens_in":13672,"feed_emoji":"⚛️","tokens_out":19761,"duration_ms":186373,"temperature":0.7,"pith_summary":"The paper proposes two quantum algorithms for least-square support vector machines. The first uses two continuous-variable modes to perform the matrix inversion at the heart of LS-SVM training, replacing many ancillary qubits and giving an evolution cost of $O(\\varepsilon^{-1}\\log(MN))$ for $M$ training samples and $N$ features. The second is a hybrid quantum-classical procedure that first compresses the large dataset into a low-rank transformed dataset using quantum singular-value tools, then solves a much smaller SVM classically, with total cost $O(N+R^3+R\\varepsilon^{-1}\\log(MN))$ where $R$ is the number of kept singular values. If these algorithms work as described, LS-SVM training and inference would depend only logarithmically on the sample count and could exploit quantum feature maps that are hard to compute classically.","feed_headline":"Quantum LS-SVM cuts training cost to log of dataset size","feed_subtitle":"A continuous-variable matrix inversion drops sample dependence to O(log(MN)); a hybrid variant adds sparse solutions.","key_machinery":"The central object is the continuous-variable representation of matrix inversion as a two-mode integral, together with the assumed factorization of the phase-estimation unitary $e^{iM_s\\hat{p}_1\\hat{p}_2} = e^{iG_1\\hat{p}_1\\hat{p}_2}e^{iG_2\\hat{p}_1\\hat{p}_2}e^{iG_3\\hat{p}_1\\hat{p}_2}$, where $G_1=\\mathrm{blkdiag}(K,0)$, $G_2=\\mathrm{blkdiag}(\\gamma^{-1}I,0)$, and $G_3=\\begin{pmatrix}0&\\mathbf{1}^T\\\\ \\mathbf{1}&0\\end{pmatrix}$. This factorization is what lets each block of the LS-SVM matrix be exponentiated separately, and it is the main source of the claimed logarithmic cost. The sparse hybrid algorithm is carried by the singular-value decomposition $\\varphi(A)=\\sum_{i=1}^{R}\\sigma_i u_i v_i^T$: QPCA and QSVT extract the top singular triples, and the paper derives reduced equations for the rotated vectors $\\alpha_{\\Omega R}$, $\\mathbf{1}_{\\Omega R}$, $y_{\\Omega R}$ and the diagonal reduced kernel $K_{\\Omega R}=\\Sigma_{R\\times R}$, so the final classical step is a small linear system.","core_discovery":"The paper's central claim is that least-square SVM can be upgraded to a simpler quantum algorithm without losing the exponential speed-up of earlier proposals. The inverse matrix $M_s^{-1}$ needed for the LS-SVM solution is generated through the continuous-variable integral $M_s^{-1} = \\frac{i}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\psi(p_1)p_2 e^{-p_2^2/2}e^{iM_s p_1 p_2}\\,dp_1\\,dp_2$, implemented by preparing a step-function state and a Gaussian state, applying the unitary $e^{iM_s\\hat{p}_1\\hat{p}_2}$, and post-selecting with homodyne detection. The paper further claims that the sample matrix can be replaced by a much smaller transformed dataset: quantum principal component analysis or quantum singular-value thresholding extracts the top $R$ singular triples of the feature matrix, and the LS-SVM parameters are then obtained by solving a reduced $R\\times R$ linear system classically. The claimed costs are $O(\\varepsilon^{-1}\\log(MN))$ for the all-quantum version and $O(N+R^3+R\\varepsilon^{-1}\\log(MN))$ for the hybrid version, both giving exponential speed-up over the number of samples.","pith_inferences":["The continuous-variable inversion gadget is a general recipe: any Hermitian matrix that can be split into efficiently exponentiating blocks could use the same two-mode integral, so the method extends beyond LS-SVM to other structured linear-algebra problems.","The hybrid pipeline implies that the practical quantum advantage is governed by the effective rank $R$ of the kernel matrix: when $R\\ll M$, the classical piece is tiny and the quantum singular-value extraction is the part that must deliver speed-up; when $R$ is large, the $O(R^3)$ classical term dominates and the advantage disappears.","A direct way to test the paper's regularization claim is to compare the classifier produced by the finite-width and homodyne inversion with the classifier produced by the regularized inverse $(M_s+\\chi I)^{-1}$ on the same synthetic dataset; the paper predicts they behave similarly."],"forward_implications":["If HVQ-SVM works as claimed, an LS-SVM training run over $M$ samples costs $O(\\varepsilon^{-1}\\log(MN))$ coherent evolution time instead of any polynomial in $M$, preserving the exponential speed-up with fewer ancillary qubits.","If QSLS-SVM works as claimed, the classical part of the hybrid pipeline solves at most an $R\\times R$ system, so for low-rank kernel matrices with $R\\ll M$ the overall method is much cheaper than classical LS-SVM.","The error analysis treats the finite squeezing width and homodyne postselection as a form of L2 regularization, so the approximation error in the continuous-variable inversion is not simply noise but a regularizing bias.","The kernel-state constructions for polynomial and radial-basis kernels mean the algorithms are intended to work with quantum feature maps, not only linear data; if such feature maps are classically hard, the proposed SVMs could classify data that classical SVMs cannot efficiently represent.","For the soft-margin SVM dual, the paper argues that the sparse solution from QSLS-SVM is one solution of the dual problem, so the sparse extraction scheme can be carried over to the standard SVM setting."],"supporting_citations":[{"why":"Supplies the original quantum LS-SVM algorithm whose matrix-inversion routine this paper is trying to simplify.","marker":"[17]"},{"why":"Supplies a quantum matrix-inversion method that the paper lists alongside HHL as the ingredient the continuous-variable routine aims to replace with fewer resources.","marker":"[22]"},{"why":"Provides the continuous-variable phase-estimation technique used to build the unitary $e^{i\\eta K\\hat p_3}$ for QSLS-SVM.","marker":"[24]"},{"why":"Provides the continuous-variable matrix-inversion integral representation and the finite-squeezing error model used in the HVQ-SVM analysis.","marker":"[25]"},{"why":"Supports the construction of the unitary $e^{iM_s\\hat p_1\\hat p_2}$ and the homodyne postselection error correction used in the algorithm.","marker":"[26]"},{"why":"Supplies quantum principal component analysis, the tool used to extract singular values and vectors in the sparse hybrid algorithm.","marker":"[27]"},{"why":"Supplies quantum singular-value thresholding, which the paper uses to keep only large singular values and to extend the method to high-rank matrices.","marker":"[29]"},{"why":"Is the quantum matrix-inversion algorithm whose resource requirements the continuous-variable method aims to reduce.","marker":"[30]"},{"why":"Gives the primal LS-SVM formulation whose reduced equations, with the top-$R$ rotated vectors, are solved by the classical part of QSLS-SVM.","marker":"[33]"},{"why":"Supplies the QRAM state-preparation cost $O(\\log(MN))$ used in the complexity claims of both algorithms.","marker":"[35]"}],"fun_headline_variants":["Quantum LS-SVM: simpler algorithm, exponential speed-up","Continuous-variable assist simplifies quantum LS-SVM","Sparse LS-SVM solutions via hybrid quantum-classical method","Quantum LS-SVM cuts cost to log, adds sparse solutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The speed-up rests on the assumption that the matrix-inversion operation can be built by applying three simple block operations in sequence at the claimed cost; the paper does not prove this sequence is valid, and the three blocks do not commute.","fun_headline_variants_meta":{"raw":{"variants":["Quantum LS-SVM: simpler algorithm, exponential speed-up","Continuous-variable assist simplifies quantum LS-SVM","Sparse LS-SVM solutions via hybrid quantum-classical method","Quantum LS-SVM cuts cost to log, adds sparse solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1530,"prompt_tokens":972,"completion_tokens":558,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":491}},"tokens_in":588,"tokens_out":558,"duration_ms":5761,"temperature":1.0,"reasoning_tokens":491,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:12:36.574685+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate Eq. (7) for a single positive eigenvalue $\\lambda$: the stated integral gives $-1/\\lambda$ rather than $\\lambda^{-1}$, which would make the inversion routine return the wrong sign. Separately, evaluate $e^{iM_s\\theta}$ against $e^{iG_1\\theta}e^{iG_2\\theta}e^{iG_3\\theta}$ for a small noncommuting kernel matrix; the difference is nonzero, and its size is the missing approximation cost in the claimed complexity.","supporting_citations":[{"cited_title":"Rebentrost, A","cited_arxiv_id":null,"evidence_quote":"Supplies a quantum matrix-inversion method that the paper lists alongside HHL as the ingredient the continuous-variable routine aims to replace with fewer resources."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the continuous-variable phase-estimation technique used to build the unitary $e^{i\\eta K\\hat p_3}$ for QSLS-SVM."},{"cited_title":"Zhang, Z.-Y","cited_arxiv_id":null,"evidence_quote":"Provides the continuous-variable matrix-inversion integral representation and the finite-squeezing error model used in the HVQ-SVM analysis."},{"cited_title":"Quantum algorithm for non-homogeneous linear partial differential equations","cited_arxiv_id":"1809.02622","evidence_quote":"Supports the construction of the unitary $e^{iM_s\\hat p_1\\hat p_2}$ and the homodyne postselection error correction used in the algorithm."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies quantum singular-value thresholding, which the paper uses to keep only large singular values and to extend the method to high-rank matrices."},{"cited_title":"Zhou, IEEE transactions on neural networks and learn - ing systems 27, 783 (2015)","cited_arxiv_id":null,"evidence_quote":"Gives the primal LS-SVM formulation whose reduced equations, with the top-$R$ rotated vectors, are solved by the classical part of QSLS-SVM."}],"review_version":1}