Variational Quantum Conditional Boltzmann Machines for Time-Series Forecasting: Architectures, Symmetric Hyperparameter Evaluation, and a Nonlinear Benchmark
2026-07-27 • Machine Learning
Machine Learning
AI summaryⓘ
The authors developed and tested four different forecasting models that use energy-based methods, including both classical and quantum approaches. They carefully tuned all model settings fairly and tested them on two types of data: one from financial processes and another from a standard nonlinear benchmark. Their results showed no clear advantage from the quantum models compared to the classical one, with some fully quantum models performing worse. The hybrid quantum-classical model performed about the same as the best classical model. They note that their sample size was limited, so very small improvements from quantum models cannot be ruled out.
Conditional Restricted Boltzmann Machine (CRBM)Quantum ComputingEnergy-Based ModelsContrastive DivergenceHyperparameter OptimizationGaussian ProcessNARMA-10 BenchmarkHybrid Quantum-Classical ModelsForecastingStatistical Power Analysis
Authors
Gerhard Hellstern, Danyal Maheshwari, Martin Zaefferer, Martin Braun, Tanja Döhler
Abstract
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.