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REVIEW 4 major objections 5 minor 1 cited by

QuLTSF: Long-Term Time Series Forecasting with Quantum Machine Learning

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A hybrid quantum-classical model called QuLTSF is claimed to beat classical linear and transformer baselines on the Weather benchmark for long-term time series forecasting.

desk verdict First QML shot at LTSF with public code, but the 'wins in all 4 cases' claim rests on borrowed baselines and single-run numbers. read the letter →

arxiv 2412.13769 v2 pith:W6E4JXHZ submitted 2024-12-18 quant-ph cs.AIcs.LG

Hari Hara Suthan Chittoor , Paul Robert Griffin , Ariel Neufeld , Jayne Thompson , Mile Gu This is my paper · ORCID
classification quant-phcs.AIcs.LG
keywords long-termtimeseriesforecastingquantummachinelearningvariationalcircuithybridquantum-classicalmodelweatherdatasetamplitudeembeddinglinearbaselines
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that adding a variational quantum circuit as a hidden layer to a simple linear forecasting model improves long-term time series forecasting. On the widely used Weather dataset, the proposed QuLTSF model reports lower mean squared error and mean absolute error than the classical linear models Linear, NLinear, and DLinear, as well as several transformer-based models, across four prediction horizons. If the claim holds, it would show that quantum machine learning can add practical value in a forecasting task where simple linear models had already outperformed much more complex transformers. The authors position this as the first application of quantum machine learning to long-term time series forecasting.

What carries the argument

The central object is the QuLTSF architecture itself: a hybrid quantum-classical model in which a variational quantum circuit (a hardware-efficient ansatz composed of trainable single-qubit gates and fixed entangling CNOT gates) sits between two classical linear layers. Amplitude embedding converts the classical linear layer's output into a quantum state, and the expectation values of Pauli-Z observables on each qubit form the quantum layer's output, which a final linear layer maps to the forecast. This design directly extends the simple Linear model of Zeng et al. by inserting a parameterized quantum feature map, and all parameters of the quantum and classical layers are trained jointly with an Adam optimizer to minimize MSE.

What would settle it

Re-run QuLTSF and the Linear, NLinear, and DLinear baselines on the Weather dataset using the same code pipeline, same 70/20/10 data split, same normalization, and multiple random seeds; if QuLTSF does not consistently achieve lower MSE and MAE than all three linear baselines across the four prediction horizons, the paper's central claim fails.

Watch

Extended reading notes

Core claim

QuLTSF is a simple three-part architecture: a classical input layer maps each univariate series of length L to a 2N-dimensional vector, amplitude embedding encodes that vector into the state of N=10 qubits, a hardware-efficient ansatz with K=3 layers applies parametrized single-qubit rotations and fixed CNOT entangling gates, and the Pauli-Z expectation values are passed through a classical output layer to produce the T-step forecast. With look-back length L=336 and prediction lengths T in {96, 192, 336, 720}, the authors report that QuLTSF outperforms all six baselines—Linear, NLinear, DLinear, FEDformer, Autoformer, and Informer—in every setting. Additional experiments with varying sequence lengths for T=96 and T=720 show QuLTSF consistently beating the three classical linear baselines. The paper also notes that in two settings QuLTSF is comparable to or slightly better than the more recent PatchTST and MTST transformer models.

Load-bearing premise

The comparison assumes that the baseline numbers quoted from Zeng et al. (2023) were produced under the same data split, normalization, and evaluation protocol used for QuLTSF, since the baselines were not re-run by the authors.

Editorial extensions

If this is right

  • If the reported results are correct, a relatively small 10-qubit variational circuit can improve forecasting accuracy over purely classical linear models on at least one standard benchmark.
  • The claimed improvements appear across all tested prediction lengths, suggesting the benefit is not limited to one specific horizon.
  • QuLTSF's performance being comparable to or better than PatchTST and MTST in limited settings suggests a quantum hidden layer may be competitive with more elaborate classical preprocessing and attention architectures.
  • The paper opens the possibility of applying hybrid quantum-classical models more broadly to long-term forecasting problems in weather, finance, and other domains.

Reading between the lines

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

  • Editorial inference: The paper does not report error bars or multiple-seed statistics, so a natural next step is to re-run QuLTSF and the classical baselines under an identical preprocessing pipeline to see whether the MSE gap is stable or within run-to-run noise.
  • Editorial inference: If the quantum layer's benefit is real, the likely mechanism is that the variational circuit supplies a rich nonlinear feature map that the purely linear model lacks; this suggests testing the same architecture on other high-dimensional regression tasks where linear baselines are strong.
  • Editorial inference: The choice of ansatz, the number of qubits, and the number of circuit layers may strongly affect the result, so a systematic ansatz search over the Weather dataset could reveal whether the reported advantage is tied to this specific circuit or generalizes across circuit designs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes QuLTSF, a hybrid quantum-classical model for long-term time series forecasting (LTSF), in which a classical linear layer maps the input to a 2N-dimensional vector, a 10-qubit hardware-efficient ansatz with K=3 layers processes an amplitude-encoded state, and a final classical linear layer outputs the T-step forecast. The authors evaluate QuLTSF on the Weather dataset with a fixed look-back length L=336 and prediction lengths T in {96,192,336,720}, and report that it outperforms Linear, NLinear, DLinear, FEDformer, Autoformer, and Informer on both MSE and MAE in all four settings (Section 5.5, Table 1). They also report experiments with varying sequence length L (Figures 2-3). The central claim is that inserting a variational quantum layer into a simple linear forecasting model yields consistent gains over classical linear and transformer baselines.

Significance. If the empirical claim were established, the paper would provide evidence that a variational quantum circuit can improve a classical linear forecasting pipeline, which is a useful contribution to the emerging QML-for-time-series literature. The model itself is transparent and easy to reproduce, the equations in Section 4 are coherent, and the authors provide public code, which are strengths. However, the significance is currently limited by the evidence base: the evaluation uses a single dataset, a single run per configuration, and baseline numbers quoted from prior work rather than recomputed under the same pipeline. Because several of the reported margins are small (e.g., T=720 MSE 0.315 vs 0.323 for DLinear; MAE 0.346 vs 0.348 for NLinear), the paper's central claim is not yet supported at the standard expected for a 'state-of-the-art' comparison. The architecture and training procedure are sound, but the empirical validation needs substantial strengthening before the superiority claim can be accepted.

major comments (4)
  1. [Table 1, Section 5.5] The headline claim that "QuLTSF outperform all the baseline models in all 4 cases" rests on a comparison between QuLTSF results produced by the authors and baseline numbers quoted from (Zeng et al., 2023), as stated in the Table 1 footnote. The paper does not specify the normalization, scaling, or exact train/validation/test split boundaries used for QuLTSF; Section 5.1 only gives a 70/20/10 split. If the QuLTSF preprocessing pipeline differs from that used to generate the quoted baseline numbers, the MSE and MAE values are not comparable even when L and T match. The authors should re-run all baselines in their own pipeline, or at minimum demonstrate that their preprocessing and split are identical to those in the source of the quoted numbers.
  2. [Section 5.4, Table 1] No error bars, confidence intervals, or repeated-seed statistics are reported. Section 5.4 lists hyperparameters (N=10, K=3, batch size 16, learning rate 0.0001) but does not state how many random seeds or initializations were used. This matters because several of the reported gains are small: at T=720, QuLTSF's MSE is 0.315 versus DLinear's 0.323, and its MAE is 0.346 versus NLinear's 0.348. Without uncertainty quantification, the observed differences can easily be within seed-to-seed or optimization-run variation, so the claim of consistent superiority is not established. The authors should provide multiple-seed results with mean and standard deviation, and ideally a significance test.
  3. [Section 5.6] The discussion introduces comparisons to PatchTST and MTST that qualify the earlier universal-superiority claim. For L=336, T=336, the paper reports QuLTSF MSE 0.248 versus PatchTST 0.249 and MTST 0.246; QuLTSF is not the best among these. The text says QuLTSF is "outperforms or comparable" to these models, but this contradicts the unqualified wording in Section 5.5 that QuLTSF outperforms all baselines in all cases. The authors should state the precise scope of the claim (e.g., limited to the baselines in Table 1) and ensure that comparisons to models outside Table 1 are accompanied by the same comparability evidence as the main table.
  4. [Section 4, Equations (2)-(4); Section 6] The conclusion states that "incorporating a hidden quantum layer yielded additional improvements," but no ablation is provided that isolates the effect of the quantum layer. QuLTSF contains trainable parameters in the input linear layer, the VQC, and the output linear layer, whereas the Linear baseline is a single linear map. It is therefore possible that any measured improvement (if real) comes from additional model capacity or the nonlinearity introduced by the input projection, rather than from the quantum circuit per se. A controlled comparison against a purely classical model with a comparable nonlinear hidden layer and a similar parameter count is needed to support the attribution in the conclusion.
minor comments (5)
  1. [Section 1] There are several typographical errors: "import problem" should be "important problem," and "strain's the models predictive capacity" should be "strains the model's predictive capacity." These should be corrected in revision.
  2. [Figures 2 and 3] The figures plot MSE for varying L but do not show error bars, markers, or legend entries that distinguish repeated runs. It is unclear whether each curve is a single run, and the y-axis ranges make the small differences between curves difficult to assess. Adding standard-deviation bands or at least markers and a clear caption would help.
  3. [Section 5.1] The abstract says "extensive experiments on a widely used weather dataset," but only one dataset is used. Either the wording should be softened to reflect the single-dataset evaluation, or additional datasets should be added.
  4. [References] Some reference names are typeset with spacing artifacts, e.g., "V aswani" for Vaswani and "Y ou" for You. These formatting issues should be fixed in the final manuscript.
  5. [Section 5.6] When citing numbers from (Zhang et al., 2024) for PatchTST and MTST, the text says "Table 2 in (Zhang et al., 2024)" but the reader must cross-check Table 1 of this paper. It would be clearer to include these comparisons in a dedicated table with the same provenance and preprocessing details as Table 1.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; only a minor background self-citation appears in the paper.

full rationale

The paper's central claim is empirical: QuLTSF is trained on the training portion of the Weather dataset and evaluated on the held-out test portion, so no prediction is derived from its own inputs by construction. The forward model in Eqs. (2)-(4) defines the output as a composition of a classical linear layer, a variational quantum circuit, and another linear layer; the superiority claim is a comparison of test-set MSE and MAE values, not a mathematical consequence of the model definition. The only citation involving the present authors is Binder et al. (2018), used in the introduction as background on recurrent quantum circuits; it does not justify the QuLTSF architecture or the reported benchmark results and is therefore not load-bearing. The reliance on baseline numbers taken from Zeng et al. (2023) raises a comparability risk about preprocessing and evaluation protocol, but that is an experimental-design concern, not circularity: no parameter is fitted to the test set and then renamed a prediction, and no equation reduces to itself. Accordingly, the circularity score is low.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The model introduces no new physical entities. The central claim depends on a handful of hyperparameters and on the comparability of externally sourced baseline numbers, which are the main free and assumed inputs.

free parameters (5)
  • Number of qubits N = 10
    Chosen by authors in Section 5.4 with no sensitivity analysis; central to the model capacity.
  • Number of ansatz layers K = 3
    Chosen by authors in Section 5.4; deeper circuits would change expressiveness and trainability.
  • Adam learning rate = 0.0001
    Declared in Section 5.4; no schedule or sensitivity reported.
  • Batch size = 16
    Declared in Section 5.4.
  • Model weights (Win, bin, VQC parameters, Wout, bout) = Trained on Weather training split
    Standard trainable parameters optimized with MSE loss; not ad hoc but part of the model.
assumptions (4)
  • domain assumption Amplitude embedding normalizes the 2N-dimensional input vector, discarding overall scale information.
    Eq. (2) provides y1, then Section 4 states amplitude embedding without discussing normalization; scale may carry forecasting signal.
  • domain assumption Channel independence: each of the M variates is modeled separately, assuming no cross-variate information is needed.
    Problem formulation in Section 3.1 feeds each univariate series separately; this discards inter-channel correlations.
  • ad hoc to paper Baseline results from (Zeng et al., 2023) are directly comparable to QuLTSF results, implying identical preprocessing, split, and evaluation.
    Table 1 footnote says 'Other results are from (Zeng et al., 2023)'; the paper does not rerun baselines or confirm identical conditions.
  • standard math Postulates of quantum mechanics (Born rule, unitary evolution) are used without proof.
    Section 2.1 introduces qubit state and measurement; standard background.

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Cite this review

Pith. "Pith review of QuLTSF: Long-Term Time Series Forecasting with Quantum Machine Learning." pith.science (2026). https://pith.science/paper/W6E4JXHZ

@misc{pith2026241213769,
  author       = {Pith},
  title        = {Pith review of: QuLTSF: Long-Term Time Series Forecasting with Quantum Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W6E4JXHZ}},
  note         = {Machine review of arXiv:2412.13769}
}
read the original abstract

Long-term time series forecasting (LTSF) involves predicting a large number of future values of a time series based on the past values. This is an essential task in a wide range of domains including weather forecasting, stock market analysis and disease outbreak prediction. Over the decades LTSF algorithms have transitioned from statistical models to deep learning models like transformer models. Despite the complex architecture of transformer based LTSF models `Are Transformers Effective for Time Series Forecasting? (Zeng et al., 2023)' showed that simple linear models can outperform the state-of-the-art transformer based LTSF models. Recently, quantum machine learning (QML) is evolving as a domain to enhance the capabilities of classical machine learning models. In this paper we initiate the application of QML to LTSF problems by proposing QuLTSF, a simple hybrid QML model for multivariate LTSF. Through extensive experiments on a widely used weather dataset we show the advantages of QuLTSF over the state-of-the-art classical linear models, in terms of reduced mean squared error and mean absolute error.

Figures

Figures reproduced from arXiv: 2412.13769 by the authors.

Figure 1
Figure 1. QuLTSF model architecture. In this section, we propose QuLTSF a hybrid QML model for LTSF and is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. MSE comparison with fixed prediction length T = 96 and varying sequence length L ∈ {48,96,192,336,504,720} [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. MSE comparison with fixed prediction length T = 720 and varying sequence length L ∈ {48,96,192,336,504,720} [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Quantum-Classical Hybrid Framework for Multivariate Time-Series Forecasting Complexity-Fidelity Trade-offs and Limitations

    cs.IR 2026-07 reject novelty 4.0 of 10

    A trainable 6-qubit-per-channel variational circuit plus linear readout roughly matches transformer forecasters at short horizons on ETT/Weather/Electricity benchmarks, but the reported setup does not match the paper'...

Reference graph

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Reviewed August 11, 2026 · model on record in the stance chip above.