{"id":"b0cf3639-1180-4979-bce2-03f8865830a6","arxiv_id":"2411.09123","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"This paper demonstrates a quantum Gaussian process using HHL and approximate quantum compiling on IBM hardware for a 32x32 kernel, but the estimated line resistance and inductance deviate substantially from true values and the objective rescaling is not mathematically justified.","lead":"This paper replaces the matrix inversion inside a Gaussian process with a quantum algorithm (HHL), compresses the circuit with approximate quantum compiling, and runs a 32 by 32 kernel inversion on IBM quantum hardware to estimate electrical line resistance and inductance. It reports that the quantum hardware estimates are on the same decimal scale as the true values but with errors up to 113 percent for inductance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rescaling after diagonal conditioning in §IV C Step 2 is algebraically invalid for non-equal kernel diagonals, so the quantum-optimized objective is not the GP marginal likelihood; the central demonstration lacks a valid basis.","rationale":"The reader's weakest_assumption precisely identifies the invalid rescaling in §IV C Step 2. My independent check confirms it: for a diagonal scaling D, y^T(DκD)^{-1}y = (D^{-1}y)^T κ^{-1}(D^{-1}y), which is not a scalar multiple of y^T κ^{-1} y unless D is a multiple of the identity. The paper's kernel matrix necessarily has unequal diagonal entries because it combines RBF kernels for vi, vj, and ii with different noise variances and signal amplitudes, so the scalar dmin rescaling is not exact. This is not merely a cosmetic approximation: the hyperparameter optimization minimizes this surrogate objective, and R and L are read off from the optimized hyperparameters. If the objective is wrong, the reported estimates do not validate the QGP pipeline. I considered other potential issues, such as lack of code/data and the inflated errors in Table I, but those affect reproducibility and strength of the demonstration rather than the logical validity of the method. The algebraic error is the most load-bearing concern because it breaks the connection between the quantum computation and the Gaussian-process objective that the paper claims to implement. The reader's REJECT verdict is therefore appropriate, and my stress-test does not change it.","tokens_in":11327,"tokens_out":7065,"duration_ms":170395,"concrete_test":"Recompute the first term of Eq. (4) on the paper's 32-point dataset two ways: (a) the true expression y^T(K+σ_n^2 I)^{-1}y, and (b) the paper's scaled and scalar-rescaled expression dmin * y^T(DκD)^{-1}y, using the actual κ and D from the experiment. Then re-optimize R and L with the corrected objective and compare to Table I. If the corrected optimum deviates from the reported R and L beyond the stated errors, the rescaling step is the cause and the QGP objective was not the GP marginal likelihood.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IV C Step 2 defines κ' = DκD with D = diag(κ_11^{-1/2}, ..., κ_nn^{-1/2}) and states that after computing ⟨y|(κ')^{-1}|y⟩ one rescales by dmin, the minimum value of D^{-1}. Algebraically, y^T(DκD)^{-1}y = (D^{-1}y)^T κ^{-1}(D^{-1}y) = Σ_{ij} d_i^{-1} y_i (κ^{-1})_{ij} y_j d_j^{-1}. This equals dmin times y^T κ^{-1} y only if all d_i^{-1} are equal, i.e., only if all diagonal entries of κ are equal. For the 32×32 kernel built from RBF kernels and three different measurement signals with different noise variances, the diagonal entries are not equal, so the condition fails. Multiplying by a single scalar cannot undo the per-component weighting d_i^{-1} d_j^{-1}. Consequently the quantity minimized by the quantum optimizer is not the GP log marginal likelihood of Eq. (4), and the hyperparameters R and L recovered from this surrogate objective do not correspond to the claimed QGP-LPE problem. The central claim that HHL-based QGP estimates line parameters on IBM Auckland therefore rests on an unproved and generally false rescaling step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a quantum Gaussian process (QGP) for multi-output regression, replacing the kernel-matrix inversion in the GP marginal likelihood (Eq. (4)) with an HHL subroutine whose circuit depth is reduced using Approximate Quantum Compiling (AQC). The method is applied to estimating resistance R and inductance L of a short transmission line from noisy voltage and current measurements, with a 32×32 kernel matrix built from 32 training samples. The authors report results from the IBM Auckland quantum device and a noiseless simulator, compare them with classical GP, and claim the implementation of up to a 13-qubit HHL circuit for the 32×32 kernel inversion.","tokens_in":11743,"tokens_out":14703,"duration_ms":135373,"significance":"If correct, this would be a notable end-to-end NISQ demonstration of HHL-based Gaussian process training on a realistic power-systems task, and the AQC depth-reduction pipeline would be a useful engineering contribution. The paper is candid about hardware limitations and reports quantitative comparisons with classical GP, which is a strength. The work does not provide machine-checked proofs or reproducible code, and its central derivation has a gap that changes the optimized objective; the headline experimental claim is also not documented at the level needed to verify it. These issues preclude acceptance in the current form.","major_comments":[{"comment":"The rescaling step after diagonal conditioning is algebraically invalid. With κ′ = DκD and D = diag(κ_11^{-1/2}, ..., κ_nn^{-1/2}), the quadratic form computed on the conditioned matrix is y^T(κ′)^{-1}y = (D^{-1}y)^T κ^{-1}(D^{-1}y) = Σ_{ij} κ_ii^{1/2} y_i (κ^{-1})_{ij} y_j κ_jj^{1/2}. This is not a scalar multiple of y^T κ^{-1} y unless all diagonal entries of κ are equal. The stated multiplication by dmin, the smallest diagonal entry of D^{-1}, cannot compensate for the per-component weights κ_ii^{1/2}κ_jj^{1/2}. For the 32×32 kernel used in Section IV B, built from RBF kernels and three measurement signals with different noise variances, the diagonal entries are not equal. Therefore the quantity minimized during QGP optimization is not the GP log marginal likelihood of Eq. (4), and the R and L estimates in Table I do not correspond to the claimed QGP-LPE objective. This is a load-bearing error for the central demonstration.","section":"IV C, Step 2"},{"comment":"The mapping from the HHL norm readout to the quadratic form of Eq. (4) is not demonstrated. The HHL ancilla measurement produces ||x||^2 for x = A^{-1}|b⟩, i.e., ⟨b|A^{-2}|b⟩, not ⟨b|A^{-1}|b⟩. In Step 1 the manuscript asserts that ||z||^2 with z=κ^{-1}|y⟩ equals ⟨y|K|y⟩, where K=(K+σ_n^2I)^{-1}. This identity requires a specific relationship between K and κ, namely K=(κκ†)^{-1}, but the 'Schutz decomposition' is never defined and no proof is provided. Without this algebraic link, the quantity actually measured on the quantum device is not established to be the first term of Eq. (4).","section":"IV C, Step 1"},{"comment":"The headline claim that 'up to 13-qubit HHL circuit has been implemented for a 32x32 kernel matrix inversion on IBM Quantum hardware' is not supported by the experimental reporting in Section V. The section provides only the final R and L estimates in Table I and graphical predictions in Fig. 6. It does not report the number of qubits used, the number of QPE evaluation qubits, the gate counts or circuit depths after AQC, or the hardware execution parameters. A reader cannot verify that a 13-qubit circuit corresponding to the 32×32 inversion was executed, as opposed to a smaller circuit or a simulator result. This claim should be removed or accompanied by the relevant circuit and execution metadata.","section":"V (Table I) and Abstract"},{"comment":"The QPE truncation to E=8 evaluation qubits is an uncontrolled approximation. The paper states that when the condition number exceeds a limit, QPE is evaluated with only 8 qubits, but it gives no error analysis for the resulting eigenvalue estimates or for the norm ||x||^2. For the conditioned matrix with condition number near O(2^9), an 8-qubit eigenvalue register may be insufficient to resolve the small eigenvalues that dominate the inverse. The authors acknowledge in Section V that heavy approximations affect accuracy, but without a bound or a numerical study the effect on the QGP objective and on the reported line-parameter estimates is unknown. This is particularly important because the approximation interacts with the conditioning step in Major Comment 1.","section":"IV C, Step 3"}],"minor_comments":[{"comment":"The manuscript contains many typos and grammatical errors (e.g., 'algrithm', 'another another', 'challanging', 'it’s', 'Posi tive', 'F ig'); a thorough language edit is needed.","section":"Throughout"},{"comment":"The symbol K is reused for the kernel matrix, its inverse, and the conditioned matrix, and κ/κ† are never precisely defined; please introduce distinct notation.","section":"IV C, Step 1"},{"comment":"The 'Schutz decomposition' is not defined and no reference is given; if it is a standard matrix factorization, the definition and the factorization algorithm should be provided.","section":"IV C, Step 1"},{"comment":"Table I reports a single run for the quantum results; for NISQ hardware, confidence intervals or repeated-shot statistics are needed to assess the significance of the R and L estimates.","section":"V, Table I"},{"comment":"Reference [17] is incomplete (author list and title missing), and reference [19] lacks the article title; these should be completed.","section":"References"},{"comment":"Figure 6 shows prediction curves without quantitative error metrics, so the statement that the hyper-parameters are 'optimized well enough' is not supported by the figure alone.","section":"V, Fig. 6"}],"recommendation":"reject","confidential_remarks":"The decisive issue is the invalid rescaling in IV C Step 2: it changes the objective function and cannot be repaired by a local rewrite. The experimental claim in the abstract is also not verifiable from the main text. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this paper actually put a 13-qubit HHL circuit for a 32×32 kernel inversion on IBM hardware for a multi-output Gaussian process line-parameter estimator. That's a real, nontrivial engineering effort, and the AQC-based depth reduction from billions to hundreds of gates is a legitimate piece of work. If you care about NISQ demonstrations in power systems, this is one of the larger HHL runs I've seen reported.\n\nThe proposed QGP extension is a natural step from refs [15] and [16]: multi-output GP, explicit kernel inversion with HHL, and hardware results. The authors are also honest about the crude accuracy: 100 optimization iterations, heavy approximations, and they don't oversell the numbers.\n\nBut the central algebraic step does not hold up. Section IV C Step 2 replaces κ by κ' = DκD with D = diag(κ_ii^{-1/2}) and then says they \"rescale back the inner product by multiplying a scale factor dmin, where dmin is the minimum value of D^{-1}\". That's not correct for a non-scalar D. The correct identity is y^T(DκD)^{-1}y = (D^{-1}y)^T κ^{-1}(D^{-1}y) = Σ_{ij} d_i^{-1} y_i (κ^{-1})_{ij} y_j d_j^{-1}. A single scalar dmin can't undo those per-component weights unless all d_i are equal, which they aren't for a kernel built from RBFs and three signals with different noise. Even in the equal-diagonal case the factor would be dmin^2, not dmin. So the quantity the quantum optimizer minimizes is not the GP log marginal likelihood of Eq. (4). The hyperparameters R and L recovered from this surrogate do not correspond to the claimed QGP-LPE objective.\n\nThat's a load-bearing flaw, not a detail. However, it's also the kind of thing a careful revision could fix: compute the quadratic form correctly on the scaled system, or absorb the scaling into the hyperparameters. The rest of the paper—the hardware setup, the AQC circuit reduction, the comparison with Qasm and classical GP—would survive.\n\nThe cited literature is fine; the one self-citation [23] is a legitimate AQC application. No code or data is provided, which makes reproduction hard, and the theoretical speedup is overstated for a dense kernel (HHL's exponential speedup needs sparsity). Those are secondary.\n\nMy take: this is a serious but flawed prototype. I'd send it to peer review because the problem is real and the fix is tractable; a referee should demand the corrected objective and a re-run before any acceptance. As written, the demonstration doesn't validate QGP.\n\nBest.","headline":"Real 13-qubit HHL kernel inversion on hardware, but the rescaling step in §IV C makes the optimizer minimize the wrong objective—needs a major fix.","tokens_in":12194,"tokens_out":4470,"would_cite":false,"duration_ms":48605,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper demonstrates a quantum multi-output Gaussian process that uses HHL to invert a 32x32 kernel matrix and estimates power-grid line parameters on a real IBM quantum computer.","keywords":["quantum Gaussian process","HHL algorithm","kernel matrix inversion","approximate quantum compiling","line parameter estimation","power systems","NISQ hardware","multi-output Gaussian process"],"falsifier":"Take any positive-definite kernel matrix with unequal diagonal entries, compute $y^T (K + \\sigma_n^2 I)^{-1} y$ classically, then compute the same quantity with the paper's D-scaling and $d_{\\min}$ rescaling; if the two disagree by more than HHL's approximation error for a well-conditioned matrix, the rescaling step does not recover the original inner product. A concrete test is a 2x2 matrix with diagonal entries 1 and 10 and the paper's conditioning recipe.","tokens_in":11162,"feed_emoji":"⚡","tokens_out":7941,"duration_ms":69698,"temperature":0.7,"pith_summary":"This paper develops a quantum version of multi-output Gaussian process regression in which the costly kernel-matrix inversion inside GP training is carried out by the HHL quantum algorithm, with Approximate Quantum Compiling used to cut circuit depth and a diagonal scaling used to reduce the kernel condition number. The authors apply this QGP to a real power-systems task: estimating the resistance R and inductance L of a short transmission line from noisy voltage and current measurements. They report running the resulting HHL circuit, up to 13 qubits for a 32x32 kernel matrix, on the IBM Auckland quantum computer, and compare the learned parameters with outputs from a noise-free simulator and from classical GP. The result is a proof-of-concept that quantum Gaussian processes can be pushed onto current NISQ hardware for a practical engineering learning problem, even though the estimated parameters are only in the same decimal range as the true values.","feed_headline":"Quantum GP estimates grid line parameters on real quantum hardware","feed_subtitle":"HHL inverts a 32x32 kernel matrix on IBM Auckland, using only voltage and current samples to learn R and L.","key_machinery":"The load-bearing machinery is HHL as a subroutine inside GP hyper-parameter optimization: the term $y^T (K + \\sigma_n^2 I)^{-1} y$ in the negative log marginal likelihood is rewritten as a norm of the solution to a linear system and evaluated by measuring the probability that HHL's auxiliary qubit is in $|1\\rangle$. Because the raw kernel has a condition number around $O(2^{30})$, the authors add strong regularization and then scale the kernel matrix by a diagonal matrix $D = \\mathrm{diag}(\\kappa_{11}^{-1/2}, \\ldots, \\kappa_{nn}^{-1/2})$ to bring the condition number down to $O(2^9)$; the inner product is then rescaled by a single factor $d_{\\min}$, the minimum entry of $D^{-1}$. To make the circuit executable, Approximate Quantum Compiling (AQC) replaces the QPE block by a shallow approximation with a bounded number of CNOT gates, using at most $E = 8$ evaluation qubits. The method therefore rests on three interacting approximations: regularization, diagonal scaling, and approximate compiling.","core_discovery":"On the paper's own terms, the central discovery is that a Gaussian process with multiple coupled outputs can be made to run on present-day quantum hardware by replacing the single expensive inversion $(K + \\sigma_n^2 I)^{-1}$ with HHL, provided the ill-conditioned kernel is first regularized and scaled, and the quantum phase estimation section of HHL is approximated by Approximate Quantum Compiling. With this pipeline the authors implemented a 32x32 kernel inversion using up to 13 qubits on IBM Auckland and obtained R and L estimates in the same decimal range as the true values (approximately 0.089 $\\Omega$ versus 0.064 $\\Omega$ for R, and approximately $5.63 \\times 10^{-5}$ H versus $2.64 \\times 10^{-5}$ H for L), with the noise-less simulator and the noisy hardware giving close results. The paper presents this as larger than earlier HHL demonstrations on real quantum hardware, and as evidence that QGP-based line parameter estimation is feasible as a proof-of-concept.","pith_inferences":["If the diagonal-scaling rescaling is corrected or removed, the same pipeline could be tested on kernel matrices with strongly varying diagonal entries; a direct numerical comparison of the HHL-estimated quadratic form against exact classical values on such matrices would separate the conditioning approximation from the AQC approximation.","Because HHL natively returns only scalar expectations $\\langle x|M|x\\rangle$, the QGP approach is naturally suited to objective functions like the log marginal likelihood, but not to applications that require full posterior covariance samples; extending it would require a different quantum linear-algebra primitive.","The claimed speedup is asymptotic and assumes sparse, well-conditioned matrices; for dense 32x32 kernel matrices the classical $O(N^3)$ inversion is trivial, so the practical value of QGP will appear only at much larger N with sparse kernel approximations, a regime not tested here.","A testable extension is to use the same conditioning-and-AQC recipe on a kernel from a different power-system task, such as state estimation or load forecasting, to see whether the parameter errors stay within the same range as reported."],"forward_implications":["A 13-qubit HHL circuit for a 32x32 kernel inversion can be executed on a real IBM quantum device, going beyond the 5-qubit, 2x2 HHL demonstrations cited in the paper.","The QGP optimizer estimates the line parameters R and L in the same decimal range as the true values, with the noisy hardware result close to the noise-less simulator, indicating the pipeline tolerates hardware noise at this scale.","With the hyper-parameters returned by QGP, the multi-output GP models can impute voltage and current signals over a full cycle, so the trained model retains predictive value even when production hardware would not be used online.","The reported execution time is far from practical for real-world deployment, so the authors position QGP as a proof-of-concept and expect near-term quantum computers to aid classical computers for offline applications."],"supporting_citations":[{"why":"Supplies the Gaussian process formalism, including the negative log marginal likelihood and the O(N^3) cost that QGP aims to reduce.","marker":"[1]"},{"why":"The multi-output GP formulation for line parameter estimation in electrical grids that this paper adapts to the quantum setting.","marker":"[2]"},{"why":"The prior proposal to use HHL for Gaussian processes, which this paper implements with additional conditioning and compiling.","marker":"[15]"},{"why":"The earlier demonstration of HHL on a 2x2 matrix with 5 qubits on real hardware, used as the comparison baseline for the 32x32 and 13-qubit result.","marker":"[16]"},{"why":"The diagonal scaling conditioning scheme used to reduce the kernel condition number from O(2^30) to O(2^9).","marker":"[20]"},{"why":"Introduces Approximate Quantum Compiling (AQC), the depth-reduction technique used on the QPE block of HHL.","marker":"[22]"},{"why":"A prior application of AQC to a real-world problem, cited to support the practicality of AQC in the pipeline.","marker":"[23]"}],"fun_headline_variants":["Quantum GP runs on IBM hardware for grid line estimation","HHL inversion powers 13-qubit quantum GP on real hardware","Quantum machine learning estimates power grid line parameters","Multi-output quantum Gaussian process tested on IBM Auckland","Quantum GP with HHL kernel inversion on 32x32 matrix"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole demonstration depends on the assumption that the diagonal rescaling of the kernel matrix can be undone by multiplying the HHL output by a single number $d_{\\min}$, which is exact only when every diagonal entry of the kernel is equal; for real measurement kernels with varying diagonal entries the rescaling silently changes the quantity being computed.","fun_headline_variants_meta":{"raw":{"variants":["Quantum GP runs on IBM hardware for grid line estimation","HHL inversion powers 13-qubit quantum GP on real hardware","Quantum machine learning estimates power grid line parameters","Multi-output quantum Gaussian process tested on IBM Auckland","Quantum GP with HHL kernel inversion on 32x32 matrix"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000177,"raw_usage":{"total_tokens":1285,"prompt_tokens":930,"completion_tokens":355,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":276}},"tokens_in":546,"tokens_out":355,"duration_ms":4273,"temperature":1.0,"reasoning_tokens":276,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:00:14.569696+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any positive-definite kernel matrix with unequal diagonal entries, compute $y^T (K + \\sigma_n^2 I)^{-1} y$ classically, then compute the same quantity with the paper's D-scaling and $d_{\\min}$ rescaling; if the two disagree by more than HHL's approximation error for a well-conditioned matrix, the rescaling step does not recover the original inner product. A concrete test is a 2x2 matrix with diagonal entries 1 and 10 and the paper's conditioning recipe.","supporting_citations":[{"cited_title":"That is, perform the transformation |0⟩nb 7→ |b⟩nb","cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian process formalism, including the negative log marginal likelihood and the O(N^3) cost that QGP aims to reduce."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The multi-output GP formulation for line parameter estimation in electrical grids that this paper adapts to the quantum setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The prior proposal to use HHL for Gaussian processes, which this paper implements with additional conditioning and compiling."},{"cited_title":"The rest of the article is organized into five main sec- tions","cited_arxiv_id":null,"evidence_quote":"The earlier demonstration of HHL on a 2x2 matrix with 5 qubits on real hardware, used as the comparison baseline for the 32x32 and 13-qubit result."},{"cited_title":"Quantum walk speedup of backtracking algorithms,","cited_arxiv_id":null,"evidence_quote":"The diagonal scaling conditioning scheme used to reduce the kernel condition number from O(2^30) to O(2^9)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces Approximate Quantum Compiling (AQC), the depth-reduction technique used on the QPE block of HHL."},{"cited_title":"Rebentrost, M","cited_arxiv_id":null,"evidence_quote":"A prior application of AQC to a real-world problem, cited to support the practicality of AQC in the pipeline."}],"review_version":1}