Exploring Hybrid Quantum-Classical Methods for Practical Time-Series Forecasting

Faculty of Sciences; I\c{l}ja Repko (1) ((1) Centre for Quantum Computer Science; Maksims Dimitrijevs (1); M\=arti\c{n}\v{s} K\=alis (1); Technology; University of Latvia)

REVIEW 5 major objections 4 minor 2 cited by

Exploring Hybrid Quantum-Classical Methods for Practical Time-Series Forecasting

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

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Pith's one-line read This paper reports that a parameterized quantum circuit trained with COBYLA achieves the lowest test mean squared error on a synthetic sales-forecasting task, beating classical linear regression and a small neural network, while a…

desk verdict An honest but underpowered empirical note: one PQC run beats two unregularized baselines on a synthetic sales series, and the VQLS negative result is likely due to a cost-function bug, not conditioning alone. read the letter →

arxiv 2412.05615 v1 pith:ZI3ITTGD submitted 2024-12-07 quant-ph

Maksims Dimitrijevs (1) , Mārtiņš Kālis (1) , Iļja Repko (1) ((1) Centre for Quantum Computer Science , Faculty of Sciences , Technology , University of Latvia) This is my paper · ORCID

classification quant-ph MSC 68Q1262M1081P68 PACS 03.67.Lx

keywords time-seriesforecastinghybridquantum-classicalparameterizedquantumcircuitvariationallinearsolverCOBYLAL-BFGS-BmeansquarederrorNISQ

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

This paper asks whether hybrid quantum-classical methods can be practical for time-series forecasting, and tests two approaches on a synthetic monthly sales dataset. A parameterized quantum circuit (PQC) trained to predict next-month sales from a 12-month window is compared against classical linear regression and a small neural network. On the single test split used, the PQC with the COBYLA optimizer posted the lowest test mean squared error, with the PQC trained by L-BFGS-B second and both classical models higher. A second quantum approach, the Variational Quantum Linear Solver (VQLS), is also implemented but fails to produce reasonable forecasts on general series because the coefficient matrix is ill-conditioned; it only behaves on strictly growing or declining series. The paper's contribution is an early empirical benchmark suggesting PQC-based forecasting can be competitive, while cautioning that the advantage rests on one unreplicated split and on optimizer choice.

What carries the argument

The central objects are the Parameterized Quantum Circuit (PQC) used as a regression model, and the Variational Quantum Linear Solver (VQLS) used to solve the normal-equation system Aw = b for sliding-window forecasting. The PQC encodes 12 past monthly values as rotation angles on 12 qubits, applies layered entangling CNOT gates and trainable RX/RY rotations, and minimizes the squared error between the expectation value of an observable and the true next value; training is done classically with L-BFGS-B or COBYLA. The VQLS approach decomposes the matrix A = X^T X into a weighted sum of tensor products of Pauli operators, prepares a normalized |b>, and uses a Hadamard test with a parameterized ansatz to minimize 1 - |<b|M|x(φ)>|^2. The paper's key finding is that the PQC machinery works well enough on this small problem to outperform the classical baselines on test MSE, while the VQLS machinery breaks down because A is ill-conditioned and the Pauli decomposition has O($n^{2}$) terms.

What would settle it

Run the same four models on the same dataset with, say, 50 different random initializations and several different train/test split points; compute the distribution of test MSE for each model. If the PQC with COBYLA does not have the lowest median test MSE, or if the differences are within the spread of the distributions, the paper's central claim that the PQC outperformed the classical baselines on this task is refuted.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that a parameterized quantum circuit with a COBYLA optimizer achieves a test-set MSE of 0.02106 on this synthetic sales-forecasting task, beating the classical linear model (0.05177) and a two-layer neural network (0.05767); a PQC with L-BFGS-B achieves 0.04418. The authors state that the PQC with COBYLA performed best on the test set, followed by PQC with L-BFGS-B. They also report that the VQLS linear-solver approach is not viable for general time-series matrices because the high condition number and the O($n^{2}$) number of Pauli terms make it ineffective; it only produces sensible extrapolations for strictly monotonic series. The paper frames these results as evidence that hybrid quantum-classical forecasting 'shows promise' but with heavy caveats about regularization, hyperparameter tuning, and the unreplicated experimental design.

Load-bearing premise

The load-bearing premise is that the single train/test split of the synthetic sales series is representative, so the observed test-MSE ordering is not a statistical fluke of initialization or the particular test period.

Editorial extensions

If this is right

If PQC-based forecasting is genuinely competitive, quantum models could be integrated into ensembles of classical forecasters to diversify errors and provide regularization.
The success of COBYLA over L-BFGS-B suggests optimizer choice is a dominant factor in PQC training outcomes; future work should systematically compare optimizers.
The VQLS failure on ill-conditioned matrices implies that quantum linear-solver approaches need preconditioning or orthogonalization (e.g., Gram-Schmidt) before they can be applied to time-series data.
The neural network's severe overfitting (training MSE 0.00003 vs test MSE 0.05767) indicates that simple classical baselines with proper regularization would likely close or invert the apparent quantum advantage.
The work motivates larger-scale benchmarks with repeated cross-validation to establish whether the observed test-MSE ordering is stable.

Reading between the lines

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

The reported advantage is fragile: because the comparison rests on a single train/test split and no repeated runs, the probability that the ordering is due to initialization luck or a particular test period is not quantified; a reasonable extension would be to run 10–50 random seeds and report median and quartiles.
The neural network's overfitting suggests that a properly regularized classical model (early stopping, dropout, or weight decay) could erase the quantum advantage entirely, so the headline claim 'PQC beats classical' is better read as 'an unregularized small NN can overfit.'
The VQLS observation that monotonic series work while realistic series fail suggests a testable threshold: measure the condition number of the sliding-window matrix for various series and see if VQLS error correlates with condition number.
A practical next step is to combine the PQC with classical post-processing (e.g., residual correction) to see if the hybrid model's errors are complementary to classical models' errors.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is a small, honest empirical note. The authors train a PQC on a synthetic monthly sales series and report a lower test MSE than a linear model and a small neural network. They also try VQLS and report that it fails. The PQC result is the kind of thing people in this area want to see, but it rests on a single unreplicated split with no repeated runs and no error bars. The paper itself admits the classical baselines are unregularized and would likely improve.

What it does well: it is clearly written, the preprocessing (differencing, scaling) is sensible, and it reports the negative VQLS result rather than burying it. The candor about the baselines and the overfitting of the neural network is genuine. That honesty is worth something.

The bigger problem is in the VQLS section. The cost function is defined as C = ⟨ψ|ψ⟩ − |⟨b|ψ⟩|² with |ψ⟩ = M|x(φ)⟩, and the authors then say |ψ⟩ is a unit vector so the first term simplifies to 1. That is not true. M|x⟩ is not normalized. The correct VQLS cost includes a denominator ⟨x|M†M|x⟩. Omitting it lets the cost be trivially minimized by shrinking the norm of x, which likely explains the 'indistinguishable from random' results. The paper blames the high condition number, but the cost bug is the more probable culprit. This should be corrected before any conclusion about VQLS is drawn.

The main PQC comparison also needs repetition. The difference between 0.021 and 0.052 on a short autocorrelated series could flip with another random initialization or a different test window. No seeds are reported. That is a real limitation, and it cuts against the 'potential advantage' phrasing.

Who is this for? Someone curious about whether PQC training behaves as advertised on small time series. It is not a breakthrough, and it does not claim to be. I would send it to a serious referee because the question is legitimate and the authors are transparent, but the VQLS section needs a fix and the PQC claim needs repeated runs. If the authors can show the gap survives multiple runs and correct the cost, this could become a useful short report.

Referee Report

5 major / 4 minor

Summary. The manuscript explores two hybrid quantum-classical methods for time-series forecasting on a synthetic monthly sales dataset. The first method trains a parameterized quantum circuit (PQC) to minimize MSE, using COBYLA and L-BFGS-B optimizers, and compares its test MSE with a classical linear regression and a small neural network. The second method applies the Variational Quantum Linear Solver (VQLS) to the normal equations from a sliding-window regression. On a single train/test split, the PQC with COBYLA reports the lowest test MSE (0.02106), followed by PQC with L-BFGS-B (0.04418), linear regression (0.05177), and the neural network (0.05767). VQLS is reported to give unsatisfactory results on the general synthetic series and is only qualitatively described as tracking monotonic trends. The authors conclude that the hybrid quantum-classical approach shows promise while acknowledging that the classical baselines lack regularization.

Significance. If the reported ordering of test MSE were robust, the result would be a modest but useful data point for NISQ-based forecasting, particularly because the PQC is trained in simulation on a realistic (though synthetic) sales series. The manuscript is honest in reporting the failure of VQLS on the general dataset and in stating that regularized classical models would likely perform better. However, the central positive claim currently rests on one unreplicated train/test split, on weak unregularized baselines, and on an incomplete experimental description. The paper does not supply code, data, seeds, error bars, or repeated trials, and the VQLS derivation contains a normalization inconsistency. These issues limit the strength of the evidence but are addressable in a revision.

major comments (5)

[Section 3, Table 1] The test-set MSE ordering in Table 1 is the sole evidence for the claim of a 'potential advantage', yet the experiments are performed on a single train/test split with no repeated runs, no seeds, and no error bars. Both PQC optimizers start from random parameters, so the reported values, including the gap between COBYLA (0.02106) and the linear model (0.05177), could be due to initialization luck or to the particular test window shown in Figure 1. The authors should run many repetitions with different seeds and initializations, report the distribution or at least the mean and confidence interval of the test MSE, and state the exact split and seeds used.
[Section 3, Table 1 and text after it] The comparison uses an unregularized classical linear model and a small neural network with no early stopping, dropout, or other regularization, and the text itself concedes that these models 'with proper regularization would likely lead to good test set MSE scores'. An advantage over deliberately weak baselines is not evidence of quantum utility. The revision should include properly regularized classical baselines, such as ridge regression, ARIMA/ETS, or a neural network with early stopping, with hyperparameters selected on the training set, and report their test MSE under the same protocol.
[Section 2.1 and Figure 2] The PQC experiment is not described with enough detail to be reproduced or evaluated: the number of qubits, ansatz depth, parameter initialization strategy, optimizer stopping criteria, learning rate for L-BFGS-B, and COBYLA iteration limits are not stated. More importantly, the text says the model minimizes the squared difference between an observable's expectation value and the labels, but it does not specify which observable is measured or how expectation values are mapped to real-valued forecasts on the original scale. These details are load-bearing because the entire positive result depends on the PQC implementation.
[Section 2.3.4, Cost Function] The cost function derivation is inconsistent: the text defines |ψ⟩ = M|x(ϕ)⟩, then writes C = ⟨ψ|ψ⟩ − |⟨b|ψ⟩|² and asserts that the first term simplifies to 1 because |ψ⟩ is a unit vector. However, M|x(ϕ)⟩ is not unit-norm in general, so C does not reduce to 1 − |⟨b|ψ⟩|². The authors must either normalize |ψ⟩ or use the correct unnormalized cost, and they should check whether this affects the reported VQLS results and the rescaling formula w ≈ (‖b‖/‖A|w⟩‖)|w⟩.
[Section 3, VQLS paragraph] The failure of VQLS is attributed to an 'extremely high condition number' and to the cost of the Pauli decomposition, but no condition number, matrix size, or numerical result for VQLS is reported. The reader cannot assess whether the failure stems from the ill-conditioning of A, from the normalization error in the cost function, or from the optimizer. The authors should report the condition number of A for the tested window sizes, the VQLS test MSE or a comparison against random predictions, and, ideally, the same quantities for the monotonic and geometric time series that are said to work.

minor comments (4)

[Figure 1 caption] The caption states that the shaded area is used for both training and scaling adjustments; please clarify explicitly that the scaling constants are computed using only the training portion and are then applied to the test portion, since the final sentence of the caption indicates test data may fall outside the scaled range.
[Section 2.3.3] The sentence 'We did VQLS experiments with 2, 3, and 4 qubits, that represents time series of length n of 4, 9, and 16' is unclear about the relationship between the qubit count, the series length, and the window size m used in the normal equations; please define the matrix dimension explicitly.
[Throughout] The text contains inconsistencies in names and formatting, such as 'Forse AI' versus 'Force AI', the author name 'M¯ arti¸nˇs K¯ alis', and the omitted circuit measurement details in Figure 4; the manuscript would benefit from a careful proofreading pass.
[Section 2.3.1] The statement that 'in reality, we need at most n^2 non-zero terms of M' is attributed to a reference about teleportation and dense coding, which is not obviously the source of this linear-algebra fact; please provide a more direct citation or a derivation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PQC and VQLS results are evaluated on a held-out test period, and no fitted quantity is defined in terms of the claimed prediction.

full rationale

The paper's central results are ordinary supervised-learning outcomes. Section 2.1 states that the PQC parameters are trained by minimizing the squared difference between the observable expectation and the actual training labels, and Table 1 reports MSE on a separate test period shown in Figure 1a. The test-set values therefore are not used to define the training objective, so the reported 'prediction' is not constructed from the target. The VQLS route is also non-circular: Section 2.2 derives the normal equations Aw = b from the training window matrix, and Section 2.3.2 rescales the normalized quantum solution by ||b||/||A|w>|| to recover w. That rescaling is a mathematical consequence of the linear-system setup, not a fit to the test output. No parameter is fitted to the test set, and no quantity is defined in terms of the result it is supposed to predict. The references are external and none is a load-bearing self-citation. The absence of repeated runs and error bars is a legitimate robustness concern about the strength of the empirical claim, but it is not circularity because the test data remain out-of-sample. Thus the derivation chain is self-contained and there are no circular steps.

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

The central claim depends on hand-picked model hyperparameters and assumptions about the dataset and quantum algorithm behavior. No new physical entities are introduced, and the only external inputs are standard linear algebra and the cited VQLS framework.

free parameters (4)

Window size m = 12
Chosen to cover a full year; hand-picked, not tuned.
Data scaling range = [-0.25, +0.25]
Chosen for quantum encoding; hand-picked without justification.
PQC ansatz depth/layers = 2 layers, 12 qubits
No justification for this architecture; hand-picked.
VQLS qubit count = 2, 3, 4
Corresponds to series lengths 4, 9, 16; chosen by experiment size.

assumptions (4)

standard math The sliding-window linear model and normal equation give a valid regression formulation.
Section 2.2, standard least squares.
domain assumption The Pauli decomposition of A can be efficiently computed and the VQLS cost function is correct.
Relies on refs [3,5,6]; assumes the quantum algorithm works as described.
domain assumption The synthetic dataset from Forse AI mimics real sales data patterns.
Section 2 states it is synthetic; no analysis of fidelity to real data.
domain assumption Differencing and scaling to [-0.25,0.25] make the series stationary and suitable for encoding.
Section 2; stationarity is asserted but not tested.

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Cite this review

Pith. "Pith review of Exploring Hybrid Quantum-Classical Methods for Practical Time-Series Forecasting." pith.science (2026). https://pith.science/paper/ZI3ITTGD

@misc{pith2026241205615,
  author       = {Pith},
  title        = {Pith review of: Exploring Hybrid Quantum-Classical Methods for Practical Time-Series Forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZI3ITTGD}},
  note         = {Machine review of arXiv:2412.05615}
}

read the original abstract

Time-series forecasting is essential for strategic planning and resource allocation. In this work, we explore two quantum-based approaches for time-series forecasting. The first approach utilizes a Parameterized Quantum Circuit (PQC) model. The second approach employs Variational Quantum Linear Regression (VQLS), enabling time-series forecasting by encoding the problem as a system of linear equations, which is then solved using quantum optimization techniques. We compare the results of these two methods to evaluate their effectiveness and potential advantages for practical forecasting applications.

Figures

Figures reproduced from arXiv: 2412.05615 by the authors.

**Figure 2.** Quantum Circuit with parameterized rotation gates. The fi [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Quantum Circuit for parameterized ansatz with 2 qubits ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Hadamard Test circuit 8 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Quantum Circuit U for the Hadamard Test, shown in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Loss function (MSE) comparison of different optimization m [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Comparison of different model predictions of the month-o [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

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'...
A generalized variational quantum linear solver on photonic platform
quant-ph 2026-07 conditional novelty 4.0 of 10

A photonic two-qubit experiment runs VQLS on 4D linear systems, adding Tikhonov regularization for singular cases and a modulo-2 cost function for finite-field equations.

Reference graph

Works this paper leans on

6 extracted references · 2 canonical work pages · cited by 2 Pith papers

[1]

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doi:10.22331/q-2018-08-06-79 2018
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Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. “Quantum Alg o- rithm for Linear Systems of Equations”. In: Phys. Rev. Lett. 103 (15 Oct. 2009), p. 150502. doi: 10.1103/PhysRevLett.103.150502

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doi:10.22331/q-2023-11-22-1188 2023
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Marcello Benedetti et al. “Parameterized quantum circuits as ma chine learning models”. In: Quantum Science and Technology 4.4 (2019), p. 043001

work page 2019
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work page doi:10.1103/physreva.70.022312 2004
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R F Werner. “All teleportation and dense coding schemes”. In: Journal of Physics A: Mathematical and General (2001). doi: 10.1088/0305-4470/34/35/332. 12

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Pith tools

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

[1] [1]

Quantum Computing in the NISQ era and beyond

John Preskill. “Quantum Computing in the NISQ era and beyond”. I n: Quantum 2 (Aug. 2018), p. 79. doi: 10.22331/q-2018-08-06-79

doi:10.22331/q-2018-08-06-79 2018

[2] [2]

Quantum Alg o- rithm for Linear Systems of Equations

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. “Quantum Alg o- rithm for Linear Systems of Equations”. In: Phys. Rev. Lett. 103 (15 Oct. 2009), p. 150502. doi: 10.1103/PhysRevLett.103.150502

doi:10.1103/physrevlett.103.150502 2009

[3] [3]

Variational Quantum Linear Solver

Carlos Bravo-Prieto et al. “Variational Quantum Linear Solver”. In: Quan- tum 7 (Nov. 2023), p. 1188. doi: 10.22331/q-2023-11-22-1188

doi:10.22331/q-2023-11-22-1188 2023

[4] [4]

Parameterized quantum circuits as ma chine learning models

Marcello Benedetti et al. “Parameterized quantum circuits as ma chine learning models”. In: Quantum Science and Technology 4.4 (2019), p. 043001

work page 2019

[5] [5]

A probabilistic and information theoretic inter- pretation of quantum evolutions

B. Reznik J. Oppenheim. “A probabilistic and information theoretic inter- pretation of quantum evolutions”. In: Quantum Physics (Aug. 2004). doi: 10.1103/PhysRevA.70.022312

work page doi:10.1103/physreva.70.022312 2004

[6] [6]

All teleportation and dense coding schemes

R F Werner. “All teleportation and dense coding schemes”. In: Journal of Physics A: Mathematical and General (2001). doi: 10.1088/0305-4470/34/35/332. 12

doi:10.1088/0305-4470/34/35/332 2001