In this study, we developed and evaluated four conditional energy-based forecasting architectures: a classical Gaussian-Bernoulli CRBM, a hybrid quantum-classical QCRBM, a full-register QQRBM, and a lag-feature QFeatureQRBM with complete derivations of their conditional distributions, Contrastive-Divergence gradients, and hybrid training, bridging the energy-based formulation and the implementation-level quantum computation. Unlike prior comparisons, our evaluation enforces symmetric hyperparameter optimisation: classical and quantum-specific hyperparameters receive an equally thorough grid search across thirteen structured experiments. We test on two data classes, a Gaussian-process dataset (GP) generated with real financial data and the input-driven NARMA-10 nonlinear benchmark. Across both regimes we find no systematic evidence of a quantum advantage at the available sample size: no quantum architecture improves on the best classical baseline. The fully quantum QQRBM and QFeatureQRBM are significantly worse, whereas the hybrid QCRBM is statistically indistinguishable from the strongest classical CRBM on both datasets. A power analysis bounds this null result: at n = 12 only medium-to-large effects are detectable, so small advantages cannot be excluded. An iso-parameter (matched-budget) comparison reaches the same conclusion: the classical CRBM is lowest at three of the four budgets and no CRBM-vs-QCRBM difference is significant at any budget.
In this study, we developed and evaluated four conditional energy-based forecasting architectures: a classical Gaussian-Bernoulli CRBM, a hybrid quantum-classical QCRBM, a full-register QQRBM, and a lag-feature QFeatureQRBM with complete derivations of their conditional distributions, Contrastive-Divergence gradients, and hybrid training, bridging the energy-based formulation and the implementation-level quantum computation. Unlike prior comparisons, our evaluation enforces symmetric hyperparameter optimisation: classical and quantum-specific hyperparameters receive an equally thorough grid search across thirteen structured experiments. We test on two data classes, a Gaussian-process dataset (GP) generated with real financial data and the input-driven NARMA-10 nonlinear benchmark. Across both regimes we find no systematic evidence of a quantum advantage at the available sample size: no quantum architecture improves on the best classical baseline. The fully quantum QQRBM and QFeatureQRBM are significantly worse, whereas the hybrid QCRBM is statistically indistinguishable from the strongest classical CRBM on both datasets. A power analysis bounds this null result: at n = 12 only medium-to-large effects are detectable, so small advantages cannot be excluded. An iso-parameter (matched-budget) comparison reaches the same conclusion: the classical CRBM is lowest at three of the four budgets and no CRBM-vs-QCRBM difference is significant at any budget.
Hybrid quantum-classical recurrent architectures have been proposed for capturing non-stationary, fat-tailed dynamics typical of financial time series, but most published evaluations compare a Quantum Long Short-Term Memory network only against plain Long Short-Term Memory on a single asset. This study contributes a controlled multi-baseline benchmark using four synthetic asset classes generated by calibrated Generalised Autoregressive Conditional Heteroskedasticity processes, comparing six architectures at a matched parameter budget below three hundred and fifty trainable parameters across five random seeds. The comparison set includes Long Short-Term Memory, Bidirectional Long Short-Term Memory, a compact Patch Time Series Transformer, Neural Basis Expansion Analysis, an Inverted Transformer, and a four-qubit Quantum Long Short-Term Memory with two variational blocks. Significance is established through Wilcoxon signed-rank tests, walk-forward validation across three expanding folds, depolarising-noise robustness sweeps, and a variational circuit depth ablation. The principal empirical finding is that compact classical recurrence dominates the benchmark on root mean squared error, while the quantum model exhibits the lowest seed-to-seed variance and the highest directional accuracy on the most volatile asset class.
Abstract Variational quantum machine learning algorithms have been proposed as promising tools for time series prediction, with the potential to handle complex sequential data more effectively than classical approaches. However, their practical advantage over established classical methods remains uncertain. In this work, we present a comprehensive benchmark study comparing a range of variational quantum algorithms (VQAs) and classical machine learning models for time series forecasting. We evaluate their predictive performance on three chaotic systems across 27 time series prediction tasks of varying complexity, and ensure a fair comparison through extensive hyperparameter optimization. Our results indicate that, in many cases, quantum models struggle to match the accuracy of simple classical counterparts of comparable complexity. Furthermore, we analyze the predictive performance relative to the model complexity and discuss the practical limitations of VQAs for time series forecasting.
Abstract Hybrid quantum machine learning (QML) relies on quantum variational circuits (QVCs) trained together with classical components. While gradient-based training is standard, it is costly on near-term hardware and vulnerable to barren plateaus, motivating gradient-free alternatives. However, there is limited systematic evidence comparing gradient-free methods to strong gradient baselines under matched quantum-evaluation budgets. A benchmark of 15 gradient-based, gradient-free, and hybrid optimizers is conducted on four binary classification datasets and two three-qubit QVC architectures using a unified experimental protocol under a fixed per-step quantum-evaluation budget. Across tasks, Adam is the strongest gradient-based method. Among the purely gradient-free schemes, Asexual Reproduction Optimization is the strongest optimizer; tournament-selection and elite-pool genetic algorithms (GA-Tourn, GA-Elite) also frequently match the best gradient-based configurations. Among the hybrid schemes, Gradient Lexicase Selection with Adam (GLS- Adam ) performs comparably. In contrast, the two gradient-free methods big bang–big crunch and a cross-entropy method variant often fail to improve beyond chance. Additional sensitivity analyses show that circuit depth can change the preferred optimizer, whereas finite-shot sampling budget and post-training backend-specific noise do not materially change the main conclusions. Overall, gradient-based methods remain robust defaults for hybrid QML, while a small subset of gradient-free and hybrid approaches provides viable alternatives under realistic quantum-evaluation constraints.
Energy-based models provide a natural bridge between statistical physics and machine learning by representing data through structured energy landscapes. Boltzmann machines are a particularly compelling class of such models for capturing complex interactions among latent variables, but their use in modern generative learning has been limited by the classical intractability of sampling from general (non-restricted) Boltzmann distributions. Here we develop a quantum-annealing-based framework that enables variational autoencoders with general Boltzmann priors. The framework employs three complementary annealing modes tailored to different stages of learning and deployment: diabatic quantum annealing provides unbiased Boltzmann samples for efficient training, slower annealing concentrates samples near low-energy configurations of the learned prior for unconditional generation, and conditional annealing with external fields steers the learned energy landscape toward attribute-specific regions for conditional generation and semantic editing. Using up to 2000 qubits on a D-Wave Advantage2 processor, we demonstrate stable training and high-quality generation on MNIST, Fashion-MNIST, and CelebA, achieving faster convergence and lower reconstruction loss than a Gaussian-prior VAE with the same encoder-decoder architecture. Beyond generation, the learned energy function provides out-of-distribution detection signals that add discriminative power beyond reconstruction loss. We demonstrate that these scores separate in-distribution samples from held-out digit classes in one-class MNIST experiments and improve the detection of market regime shifts in financial data. These results establish quantum annealing as a practical and controllable physical mechanism for energy-based representation learning and generative modeling beyond the reach of tractable classical approaches.
Time series forecasting is a valuable tool for many applications, such as stock price predictions, demand forecasting or logistical optimization. There are many well-established statistical and machine learning models that are used for this purpose. Recently in the field of quantum machine learning many candidate models for forecasting have been proposed, however in the absence of theoretical grounds for advantage thorough benchmarking is essential for scientific evaluation. To this end, we performed a benchmarking study using real data of various quantum models, both gate-based and annealing-based, comparing them to the state-of-the-art classical approaches, including extensive hyperparameter optimization. Overall we found that the best classical models outperformed the best quantum models. Most of the quantum models were able to achieve comparable results and for one data set two quantum models outperformed the classical ARIMA model. These results serve as a useful point of comparison for the field of forecasting with quantum machine learning.
Multi-output time-series forecasting in energy systems is challenging because of nonlinear dynamics, multi-scale seasonality, and strong dependencies across correlated series. In this work, we investigate two hybrid quantum-classical frameworks for multi-stream time-series forecasting on a real Smart Meter dataset comprising 103 household electricity consumption time-series, with experiments executed on the ibm_marrakesh superconducting quantum processor. The first model, Kernelized Quantum Reservoir Computing with Repeated Measurement (KQRC-RM), combines coupled quantum reservoirs, ancilla-assisted repeated measurement, and kernelized readouts to model temporal dynamics and cross-stream correlations jointly. For a 3-stream time-series input and output, the KQRC-RM model using 114 qubits achieves an MAE of 0.0811 on MPS simulator (36.92\% improvement over its classical analog) whereas performance degrades to an MAE of 0.1524 on hardware. The second, a Projected Quantum Kernel Gaussian Process (QGP), replaces fidelity-based kernels with projected kernels constructed from local reduced-state statistics. Using a topology-aware 100-qubit QGP model to predict 100 multi-output time-series values, we observe 49\% of time-series outputs achieve high-accuracy predictions (MAE <0.15), with an average MAE of 0.082 for this low-error group. The medium-error regime (MAE 0.15-0.35) has an average MAE of 0.229, while the high-error regime (MAE >0.35) has an average MAE of 0.664. Overall, this reduces the average MAE relative to the classical GP baseline by 62.01\% on MPS simulator and 40.37\% on hardware. Together, these results demonstrate the feasibility of hybrid quantum machine learning for multi-input, multi-output time-series forecasting at the 100+ qubit scale on NISQ devices.