{"id":"adf6ef69-f7ba-474d-9fb0-77818d0a879a","arxiv_id":"2607.06232","paper_version":1,"verdict":"CONDITIONAL","confidence":"UNKNOWN","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":5,"one_line_summary":"Two machine learning models reconstruct continuous Wigner functions from sparse phase-space measurements: a provably efficient regression model for sparse states (O(s⁴ log d) samples) and a self-supervised neural network for general states including experimental GKP code data.","lead":"The paper develops machine learning methods to reconstruct quantum Wigner functions from far fewer measurements than standard techniques. For certain quantum states, they prove measurement costs scale logarithmically with system size; for harder states like GKP error-correcting codes, a neural network reconstructs the function from sparse data and works on real experimental measurements.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The O(log d) sample complexity bounds a fixed-ε L² risk, not the fixed-fidelity criterion used in experiments; translating between these metrics introduces polynomial d-dependence because the phase-space support of cat states grows with d.","rationale":"The reader correctly identifies the gap between theoretical O(log d) and empirical O(d^0.6) scaling for cat states, but attributes it to sparsity conditions not capturing practical difficulty. I believe the issue is more fundamental: it is a metric mismatch. The theorems bound the expected L² prediction error R(cW) ≤ ε for fixed ε, while experiments threshold on fidelity F ≥ 0.99. Because the Wigner function support of cat states grows with d, translating fixed ε to fixed fidelity introduces polynomial d-dependence, making the effective sample complexity for practical reconstruction polynomial rather than logarithmic.\n\nThe proofs of Theorems 1–2 are technically correct — they use standard Lasso generalization bounds (Lemma 1 from Mohri et al.) with appropriate choices of t, r_∞, and M. The Gabor frame truncation analysis (Proposition 1) is also sound, relying on exponential decay of the STFT (Lemma 3). The concern is not with the proofs themselves but with whether the guaranteed metric (fixed-ε L² risk) corresponds to the practically relevant metric (fixed fidelity).\n\nThis concern does not invalidate the theorems — they are correct as stated. But it does mean that the headline claim of 'O(log d) measurement complexity' should be qualified: this scaling holds for fixed ε in L² risk, which corresponds to a fidelity threshold that degrades with d for states whose phase-space support grows. The practical advantage over conventional tomography is still present (the empirical O(d^0.6) is much better than O(d²)), but it is less dramatic than the theoretical O(log d) suggests.\n\nThe DNN model's lack of theoretical guarantees, which the reader also raises, is a legitimate concern but is less load-bearing because the paper is transparent about it and frames the DNN as an empirical approach. The metric mismatch concern is more significant because it affects the interpretation of the paper's central theoretical contribution.\n\nI recommend keeping the verdict as CONDITIONAL. The theoretical results are correct but their practical relevance for fixed-fidelity reconstruction needs clarification — either by showing that the O(log d) scaling translates to fixed fidelity (which would require addressing the support-area dependence) or by explicitly stating that the logarithmic scaling applies to the L² risk metric with fixed ε.","tokens_in":52184,"tokens_out":14319,"duration_ms":977948,"concrete_test":"For the cat states in Fig. 5a (varying α, hence varying d), recompute the minimum number of phase-space samples needed to achieve R(cW) ≤ ε₀ for a fixed ε₀ (e.g., 0.01) in the L² risk, instead of F ≥ 0.99. This requires evaluating E_α~D[|cW(α)−W(α)|²] on a held-out test set drawn from D. If the L²-risk-based sample complexity scales as O(log d) while the fidelity-based one scales as O(d^0.6), this confirms the metric mismatch is the source of the gap. If both scale polynomially, the gap arises from other factors (Lasso optimization, hidden constants).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorems 1–2 guarantee R(cW) = E_α~D[|cW(α) − W(α)|²] ≤ ε with N = O(s⁴ log d / ε²) samples. This is a per-point average squared error under distribution D. The experiments, however, threshold on fidelity F ≥ 0.99, where F = (π/2)∫cW·W dα. For a pure state, ∫|cW − W|² ≈ 4(1−F)/π ≤ 0.04/π when F ≥ 0.99. If D is uniform over the phase-space support region of area A, then R(cW) ≤ ε implies ∫_A|cW−W|² ≤ Aε. Achieving F ≥ 0.99 thus requires ε ≤ O(1/A). For cat states, the Wigner function support (two Gaussian peaks plus interference fringes of length ~|α|) has area A = O(|α|) = O(√d). This yields an effective sample complexity N = O(s⁴ A² log d) = O(s⁴ d log d) for fixed fidelity — polynomial, not logarithmic. This metric mismatch explains the empirical O(d^0.6) scaling for cat states (Fig. 5a): the theory's O(log d) holds for fixed ε in L² risk, which corresponds to a fidelity threshold that degrades as 1 − O(εA) = 1 − O(ε√d), not the constant F ≥ 0.99 used experimentally. The paper does not address this translation. For binomial code states (Fig. 4a), the phase-space support grows much more slowly with d, so the mismatch is less severe and the near-flat scaling is consistent with the theory.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper develops two machine learning approaches for reconstructing Wigner functions of continuous-variable quantum states from sparse pointwise phase-space measurements. The first is a sparse regression model with PAC-learning guarantees: for states that are s²-sparse in the Fock basis, Theorem 1 gives sample complexity N = O(s⁴ log d / ε²) for expected L² risk ≤ ε; for s-sparse coherent-state superpositions, Theorem 2 gives N = Õ(s⁴ κ⁻⁴ log R / ε²). The second is a self-supervised deep neural network (DNN) with an encoder-decoder architecture that learns a neural implicit representation of the Wigner function, applicable to states beyond the sparse regime (e.g., GKP states). The DNN model is validated on simulated GKP states and on experimental circuit-QED data from a beyond-break-even quantum error correction experiment, where it reconstructs Wigner functions from 27×27 sparse grids and identifies the dominant photon-addition error process. The regression proofs use standard Lasso generalization theory (Mohri et al., Theorem 11.16) combined with explicit feature maps (Laguerre polynomials for Fock sparsity, Gabor frames for coherent-state sparsity).","tokens_in":52539,"tokens_out":5883,"duration_ms":284432,"significance":"The paper makes a genuine contribution by introducing PAC-learning guarantees for Wigner function reconstruction from displaced-parity measurements, a setting not covered by prior CV state learning results that focus on homodyne/heterodyne data. The parameter-free derivations in Theorems 1–2 (with t = √2s, r∞ = 2/π, M = 4√2s/π) are clean and the sparsity-to-sample-complexity connection is well-motivated. The Gabor frame construction for coherent-state sparse support, including the truncation error analysis via Gelfand–Shilov decay (Proposition 1), is technically nontrivial. The experimental demonstration on circuit-QED GKP data—including identification of the dominant error subspace from sparse measurements—is a practically valuable result. The honest comparison between regression and DNN models (Sec. VII), acknowledging that the DNN's spectral bias causes it to underperform on oscillatory Wigner functions, is commendable.","major_comments":[{"comment":"§V.A.2, Fig. 5a, and Theorems 1–2: The theorems bound the expected L² risk R(cW) = E_{α∼D}[|cW(α)−W(α)|²] ≤ ε for fixed ε, while the experiments threshold on fidelity F ≥ 0.99 (Eq. 30–31). The paper does not address the translation between these metrics. For a pure state, 1−F ≈ (π/4)∫|cW−W|² dα. If D is uniform over the phase-space support region of area A, then R(cW) ≤ ε implies ∫|cW−W|² ≤ Aε, so achieving F ≥ 0.99 requires ε = O(1/A). For cat states, the support area A = O(|α|) = O(√d), yielding an effective sample complexity N = O(s⁴ A² log d) = O(s⁴ d log d) for fixed fidelity—polynomial, not logarithmic. This metric mismatch naturally explains the empirical O(d^{0.6}) scaling for cat states (Fig. 5a) and the near-flat scaling for binomial code states (Fig. 4a), whose support grows slowly with d. The paper should explicitly discuss this translation, clarify in the abstract and §I–IVA","section":null}],"minor_comments":[{"comment":"§IV.A.2, Eq. (192) in the proof of Theorem 2: the derived sample complexity reads N = O(s⁴/(ε²κ⁴) · log⁴(R + log(s/(κ√ε)))/(8√δ)), which appears inconsistent with the theorem statement N = Õ(s⁴/(ε²κ⁴) log R / δ). Based on the derivation via Lemma 1, the dependence should be log(d_f) = O(log(R + log(s/(κ√ε)))), not log⁴. Please verify and correct Eq. (192).","section":null},{"comment":"§IV.A.1, line after Eq. (16): the phrase 'exponential improvement in the d-dependent scaling' is slightly misleading; the improvement is from O(d²) to O(log d), which is super-polynomial but not typically described as 'exponential.' Consider rephrasing.","section":null},{"comment":"§IV.B: the DNN model's 'smooth structure' assumption is stated informally. While the paper acknowledges its limitations in §VII, a brief formal characterization (e.g., bounded Hessian or band-limitedness) would help readers understand the regime of applicability.","section":null},{"comment":"Fig. 5a: the distinction between 'total' and 'non-zero' sampled points is important but could be clearer in the figure caption. Consider explicitly defining what constitutes a 'non-zero' point (threshold on |W|).","section":null},{"comment":"§IX.B, Eq. (66): the total measurement cost includes a factor of log(s⁴/(ε²δ)) from the per-point shot count M. This is correct but the distinction between 'number of phase-space points' N and 'total number of measurement shots' M·N should be stated more prominently in the main text, as the abstract and introduction refer to 'measurement complexity' without specifying which quantity.","section":null},{"comment":"References: the paper cites arXiv:2603.18136 (2026) and arXiv:2512.14831 (2026), which appear to be future-dated. Please verify these citations.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the metric mismatch between fixed-ε L² risk and fixed fidelity is the primary issue. It is legitimate and substantive: the paper's headline claim of 'logarithmic scaling with d' is correct for fixed ε but does not directly translate to the fixed-fidelity criterion used in experiments. The authors should be asked to add a dedicated discussion of this point, as it affects how readers interpret the practical relevance of Theorems 1–2. The theoretical results themselves are sound; the issue is in the bridge between theory and experiment. I would not consider this a reason for rejection if the authors add the discussion and adjust the framing accordingly."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and the insightful observation regarding the metric mismatch between our theoretical guarantees (expected L² risk) and our experimental evaluation (fidelity). The referee's analysis is essentially correct, and we will revise the manuscript to explicitly address this translation and its implications for the scaling results.","responses":[{"response":"We thank the referee for this incisive and entirely correct observation. The metric mismatch between the theoretical guarantee (expected L² risk R(cW) ≤ ε) and the experimental evaluation metric (fidelity F ≥ 0.99) is a genuine gap in the current presentation, and the referee's analysis of the translation between these metrics is accurate.","revision_made":"yes","referee_comment":"§V.A.2, Fig. 5a, and Theorems 1–2: The theorems bound the expected L² risk R(cW) = E_{α∼D}[|cW(α)−W(α)|²] ≤ ε for fixed ε, while the experiments threshold on fidelity F ≥ 0.99 (Eq. 30–31). The paper does not address the translation between these metrics. For a pure state, 1−F ≈ (π/4)∫|cW−W|² dα. If D is uniform over the phase-space support region of area A, then R(cW) ≤ ε implies ∫|cW−W|² ≤ Aε, so achieving F ≥ 0.99 requires ε = O(1/A). For cat states, the support area A = O(|α|) = O(√d), yielding an effective sample complexity N = O(s⁴ A² log d) = O(s⁴ d log d) for fixed fidelity—polynomial, not logarithmic. This metric mismatch naturally explains the empirical O(d^{0.6}) scaling for cat states (Fig. 5a) and the near-flat scaling for binomial code states (Fig. 4a), whose support grows slowly with d. The paper should explicitly discuss this translation, clarify in the abstract and §I–IVA"},{"response":"We will incorporate the following revisions in the next version of the manuscript: (1) We will add a dedicated discussion (in §V or §VII) explicitly deriving the relationship between expected L² risk and fidelity for pure states. As the referee notes, for a pure state, 1−F ≈ (π/4)∫|cW−W|² dα, and when D is uniform over a support region of area A, the bound R(cW) ≤ ε translates to ∫|cW−W|² ≤ Aε, so achieving F ≥ 0.99 requires ε = O(1/A). This yields an effective sample complexity of N = O(s⁴ A² log d) for fixed fidelity rather than fixed ε. (2) We will clarify in the abstract, §I, and §IV.A that the O(log d) scaling in Theorems 1–2 applies to fixed expected L² risk ε, and that when the target metric is fidelity at a fixed threshold, the effective sample complexity acquires an additional factor of A², where A is the phase-space support area. For states whose support area grows with d (such as cat states, where A = O(|α|) = O(√d)), this yields an effective scaling of O(s⁴ d log d) rather than O(s⁴ log d). (3) We will explicitly note that this translation provides a natural explanation for the empirical scaling observed in Figs. 4a and 5a: the near-flat scaling for binomial code states with fixed sparsity (Fig. 4a) is consistent with the fact that their phase-space support area grows slowly with d, while the O(d^{0.6}) scaling for cat states (Fig. 5a) is consistent with the polynomial-in-d dependence introduced by the growing support area, with the sublinear exponent reflecting that the worst-case bound is not tight in practice. We note that the theoretical results themselves (Theorems 1–2) are correctly stated as bounds on expected L² risk; the issue is solely that the manuscript did not previously discuss the translation to the fidelity metric used in experiments. We are,","revision_made":"yes","referee_comment":"(continued) The paper should explicitly discuss this translation, clarify in the abstract and §I–IVA"}],"tokens_in":51956,"tokens_out":2564,"duration_ms":130239,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Bottom line: this paper introduces a genuinely new formulation — Wigner function reconstruction as a PAC learning problem with logarithmic sample complexity for sparse states — and the theory is sound. The DNN model works well in practice but has no guarantees and one metric mismatch the authors don't address.","headline":"PAC learning framework for Wigner function reconstruction from displaced-parity measurements is new and correct; the DNN model is useful but lacks theory.","tokens_in":53093,"tokens_out":125,"would_cite":true,"duration_ms":118561,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.Wj","03.67.Lx","42.50.Dv"],"model":"glm-5.2","headline":"Machine learning cuts Wigner tomography cost from d² to log d","keywords":["Wigner function tomography","continuous-variable quantum states","machine learning","sparse regression","deep neural networks","GKP states","bosonic quantum error correction","sample complexity"],"falsifier":"If a state claimed to be s²-sparse in the Fock basis required measurement complexity scaling faster than O(s⁴ log d) in practice, or if the DNN consistently failed to reconstruct GKP Wigner functions from sparse data with overlap exceeding 0.99 on experimental measurements, the central claims would be undermined.","tokens_in":52344,"feed_emoji":"","tokens_out":1406,"duration_ms":155524,"temperature":0.7,"pith_summary":"This paper develops two complementary machine learning approaches for reconstructing Wigner functions of continuous-variable quantum states from sparse pointwise phase-space measurements, replacing the dense grid of d² points required by conventional tomography. The first approach is a sparse regression model with provable guarantees: for states that are s²-sparse in the Fock basis (Theorem 1), O(s⁴ log d / ε²) measurement points suffice to achieve expected prediction error ≤ ε, and for s-sparse coherent-state superpositions (Theorem 2), Õ(s⁴ κ⁻⁴ log R / ε²) points suffice, both scaling logarithmically with the effective Hilbert-space dimension d when the sparsity s is constant. The second approach is a deep neural network with an encoder-decoder architecture that learns a continuous implicit representation of the Wigner function from coarse-grained measurement data, enabling prediction at arbitrary phase-space resolution without prior assumptions about the state's structure. The DNN is trained self-supervised on paired low- and medium-resolution resamplings of the measurement grid. The two models are complementary: the regression model excels on states with known sparsity (binomial code states, cat states) where it captures high-frequency oscillatory features, while the DNN excels on states with smooth phase-space structure (GKP states, random SNAP-displacement circuit outputs) and is robust to experimental noise. On experimental circuit-QED data of GKP code states subjected to multiple rounds of quantum error correction, the DNN reconstructs Wigner functions from one-ninth of the measurement grid, identifies the dominant error subspace as photon-addition (heating) noise, and tracks code-space fidelity and logical-state evolution across QEC cycles.","feed_headline":"Log d, not d²: ML shrinks Wigner tomography cost","feed_subtitle":"Sparse regression and deep learning reconstruct continuous-variable Wigner functions from logarithmically fewer phase-space measurements, as","key_machinery":"Sparse regression with Lasso on Laguerre-polynomial feature maps (Fock-sparse states) and Gabor-frame feature maps (coherent-state-sparse states); deep neural network with residual CNN encoder and coordinate-conditioned MLP decoder for implicit neural representation; self-supervised training on paired low/medium-resolution grid resamplings; spectral decomposition of density matrices reconstructed via MLE to extract logical and error subspaces","core_discovery":"The central mechanism is that Wigner-function reconstruction can be cast as a supervised learning problem where the sample complexity is governed not by the Hilbert-space dimension d but by structural properties of the target state. For the regression model, the key insight is that the Wigner function's linearity (Eq. 3) decomposes it into a sparse linear combination of elementary Wigner functions, enabling Lasso regression with a sample complexity bound of O(s⁴ log d) that depends on the sparsity s rather than d². The proof for Fock-sparse states uses a Laguerre-polynomial feature map with bounded sup-norm (2/π) and ℓ₁-norm constraint O(√2 s), yielding generalization error via standard Lass","pith_inferences":["The logarithmic scaling for sparse states implies that the information-theoretic bottleneck in Wigner tomography is not the Hilbert-space dimension per se but the number of independent degrees of freedom in the phase-space representation — a compressed-sensing-like separation that could extend to other quasi-probability distributions (Husimi Q, P functions).","The DNN's spectral bias toward low-frequency patterns, while a limitation for cat-state interference fringes, may be an asset for experimental data where high-frequency features are often dominated by shot noise rather than genuine quantum structure — the model may act as an implicit denoiser.","The gap between the theoretical O(log d) prediction and the empirical O(d^0.6) scaling for cat states suggests that the separation condition (Eq. 19) may not fully capture the practical difficulty of resolving closely spaced coherent-state amplitudes, and that tighter bounds incorporating phase-space resolution constraints could close this gap.","Multi-mode Wigner function reconstruction, where the dimension scales as d^(2n) for n modes, would make the log d vs d² separation even more dramatic, potentially making structure-aware learning essential rather than optional for multi-mode bosonic systems."],"forward_implications":["For experimentally relevant states with constant sparsity (Fock or coherent-state), Wigner tomography cost drops from quadratic to logarithmic in Hilbert-space dimension, potentially enabling characterization of high-photon-number bosonic states that are currently intractable.","The DNN's ability to identify dominant error processes from sparse data could become a practical diagnostic tool for bosonic quantum error correction experiments, reducing the measurement overhead of routine QEC validation.","The regression framework extends PAC learning from discrete qubit systems to continuous phase-space representations, suggesting that classical surrogates for quantum state characterization can arise whenever the target representation has exploitable low-complexity structure.","The separation between regression (optimal for oscillatory Wigner functions) and DNN (optimal for smooth Wigner functions) suggests a hybrid strategy where the model is selected based on the expected phase-space structure of the target state."],"fun_headline_variants":["ML reconstructs continuous Wigner functions from sparse data","Sample complexity for Wigner tomography scales with sparsity","Deep learning maps sparse measurements to Wigner functions","Logarithmic measurement scaling for Wigner tomography via ML","ML-based Wigner reconstruction from fewer phase-space samples"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The theoretical guarantees depend on the target state being genuinely sparse in a known basis (Fock or coherent-state), and the DNN model implicitly assumes that the Wigner function has smooth structure amenable to neural implicit representation — an assumption that fails for highly oscillatory states like binomial code states and cat-state interference fringes, where the DNN underperforms the regression model.","fun_headline_variants_meta":{"raw":{"variants":["ML reconstructs continuous Wigner functions from sparse data","Sample complexity for Wigner tomography scales with sparsity","Deep learning maps sparse measurements to Wigner functions","Logarithmic measurement scaling for Wigner tomography via ML","ML-based Wigner reconstruction from fewer phase-space samples"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1106,"prompt_tokens":549,"completion_tokens":557,"prompt_tokens_details":null},"tokens_in":549,"tokens_out":557,"duration_ms":14985,"temperature":1.0,"reasoning_tokens":535,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T12:32:49.603501+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If a state claimed to be s²-sparse in the Fock basis required measurement complexity scaling faster than O(s⁴ log d) in practice, or if the DNN consistently failed to reconstruct GKP Wigner functions from sparse data with overlap exceeding 0.99 on experimental measurements, the central claims would be undermined.","supporting_citations":[],"review_version":1}