We characterize the learning dynamics of a compact hybrid quantum forecasting model through comparison with a structurally aligned classical baseline. Using stationary harmonic-mixture and nonstationary chirp benchmarks with controlled spectral complexity and data availability, we analyze empirical Neural Tangent Kernel dynamics through kernel-target alignment, kernel drift, spectral concentration, and training loss. The classical model exhibits stronger early target alignment, whereas the hybrid model generally develops a less concentrated kernel spectrum and smaller kernel drift. Despite these distinct optimization geometries, both architectures attain similar held-out performance across the evaluated regimes. Notably, the hybrid model uses 125 trainable parameters compared with 281 for the classical baseline and reaches its validation-selected checkpoint earlier in 15 of 18 frequency conditions. A Fourier-augmented classical baseline does not reproduce the observed training behavior, while a controlled re-uploading ablation shows that repeated encoding systematically modifies both optimization and kernel geometry. These results demonstrate that comparable generalization can emerge from substantially different learning trajectories and that individual NTK diagnostics do not provide monotonic predictors of validation convergence. Rather than claiming a general quantum advantage, the study identifies architecture-dependent learning behavior that is masked by endpoint accuracy alone.
Figures & tables
Fig. 1 : NTK Dynamics for the Harmonic-mixture dataset with 200 training samples.
Fig. 2 : NTK Dynamics for the Harmonic-mixture dataset with 50 training samples.
Fig. 3 : NTK Dynamics for the Chirp dataset with 200 training samples.
Fig. 4 : NTK Dynamics for the Chirp dataset with 50 training samples.
Freq
Epoch reduction(%)
pE
Train AULC gain(%)
pA
1
11.87
0.0048
-9.44
0.0012
2
9.20
0.0185
-12.40
< 0.0001
4
26.80
0.0197
-5.35
0.0353
8
24.36
0.00316
-0.35
0.8469
12
58.73
0.0009
8.84
< 0.0001
16
-6.06
0.0990
5.72
< 0.0001
TABLE I : Frequency-dependent learning dynamics for the 50-sample harmonic-mixture benchmark. Positive epoch reduction indicates earlier attainment of the validation-selected checkpoint by the hybrid model; positive AULC gain indicates lower training AULC. pE and pA correspond to paired Wilcoxon signed-rank tests over 20 initialization seeds.
Dataset
N
Test gain ( % )
Epoch reduction
Q earlier
Harmonic
200
-2.21
4.72
5/6
Harmonic
50
-1.07
5.43
5/6
Chirp
200
1.40
11.53
3/3
Chirp
50
-3.58
3.12
2/3
TABLE II : Validation-selected convergence and held-out performance across forecasting benchmarks. Values are averaged across evaluated frequencies and 20 initialization seeds. Positive test gain indicates lower test MSE for the hybrid model; epoch reduction indicates earlier attainment of the validation-selected checkpoint. The last column indicates in how many frequency the quantum model achieved the checkpoint earlier
Fig. 5 : Effect of the number of data re-uploading operations on final loss and empirical NTK metrics for the harmonic-mixture benchmark with 50 training samples. Circuit depth is fixed to four rotation–entanglement layers.
Time series forecasting largely benefits from combining the strengths of different models, especially using a scheme where a model corrects another model by capturing supplementary patterns from forecasting errors. Concurrently, quantum models are providing a means to augment the classical capacity, including in time series forecasting, by acting alongside classical models in hybrid architectures. In this work, we propose the first forecasting system based on error correction that jointly uses quantum and classical models. Here, quantum models first extract patterns by exploring quantum phenomena, and classical models capture the remaining patterns from the quantum errors. Compared to classical single models and classical-classical hybrid models based on error correction, the complementary capacity that emerges from this quantum-classical system provided the best results in most of the addressed problems. Therefore, this work paves the way to introduce quantum models in established hybridization schemes for time series forecasting.
Jonathan H. A. de Carvalho, Filipe C. de L. Duarte, Fernando M. de Paula Neto +1
Centro de Inform´atica, Universidade Federal de Pernambuco, Recife, Brazil · Departamento de Finan¸cas e Contabilidade, Universidade Federal da Para´ıba, Jo˜ao Pessoa, Brazil
This paper presents a unified quantum-classical hybrid framework for multi-horizon time-series forecasting, introducing two model variants Quantum Reservoir Forecaster (QRC-F) and Variational Quantum Forecaster (VQF-F). The proposed framework investigates the complexity-fidelity trade-off of quantum forecasting under near-term NISQ hardware constraints. Continuous time-series signals are transformed into binary representations through uniform quantization and encoded into quantum states using angle encoding with parameterized RY rotation gates. Cross-channel entanglement layers capture dependencies among multiple variables. QRC-F utilizes a fixed random unitary quantum reservoir for stable, gradient-free temporal feature extraction, whereas VQF-F employs a trainable variational quantum circuit optimized through the parameter-shift rule to learn temporal and inter-variable patterns from Pauli expectation values. Both models replace computationally expensive quadratic self-attention with efficient linear transformations, reducing parameter complexity. A shared MIMO-based multi-horizon prediction head simultaneously generates forecasts across multiple horizons, avoiding error accumulation in recursive forecasting. Experimental evaluations on benchmark datasets, including ETTh1, ETTh2, ETTm1, ETTm2, Weather, electricity, and exchange-rate, demonstrate that VQF-F achieves superior training stability and parameter efficiency, while QRC-F provides enhanced robustness and circuit fidelity under quantum noise. The results establish a practical quantum-native forecasting framework with strong potential for deployment on near-term NISQ devices.
Sanjay Chakraborty, Fredrik Heintz
Department of Computer Science & Engineering, Techno International New Town, Kolkata, India
We evaluate hybrid quantum-classical machine learning for the reduced-order prediction of spatiotemporal brain deformation fields. To mitigate the computational intractability of high-dimensional displacement fields, we employ Proper Orthogonal Decomposition (POD) to project the data into a compact latent space. Within this framework, we formulate two distinct learning objectives: static temporal-to-latent regression and autoregressive latent state forecasting. We systematically benchmark compact classical baselines against both minimal and enhanced hybrid quantum architectures. Our results demonstrate that classical networks provide the strongest baselines in the present setting. For static regression, a classical POD-MLP outperforms all evaluated quantum variants, although an enhanced Variational Quantum Circuit (VQC) substantially improves upon a minimal VQC baseline. For temporal forecasting, a classical POD-LSTM delivers superior predictive accuracy and statistical robustness compared to an enhanced Quantum LSTM (QLSTM) across varying history windows and random initializations. Overall, this study establishes reduced-order physical field learning as a rigorous testbed for near-term QML, highlighting that while hybrid enhancements successfully recover expressivity in weak quantum circuits, classical architectures retain a definitive advantage in both fidelity and stability.
Tao Liu, Ge He, Dongyu Liang +1
Lawrence Technological University · North Carolina State University