Variational Quantum Conditional Boltzmann Machines for Time-Series Forecasting: Architectures, Symmetric Hyperparameter Evaluation, and a Nonlinear Benchmark

2026-07-27

AI Summary

This study investigates whether quantum computing can surpass classical conditional restricted Boltzmann machines (CRBMs) in time series forecasting and examines the existence of a quantum advantage under fair hyperparameter configurations. To this end, the authors construct and uniformly evaluate four conditional energy-based models: the classical CRBM, a hybrid quantum-classical QCRBM, a fully quantum QQRBM, and a QFeatureQRBM incorporating lagged features. They derive, for the first time, the conditional distributions and training gradients for these models and employ a symmetric hyperparameter grid search to ensure equitable comparison. Experiments on Gaussian process and NARMA-10 benchmarks reveal no significant performance difference between the hybrid QCRBM and the classical CRBM, while fully quantum models perform worse. No systematic quantum advantage is observed, though a marginal benefit cannot be ruled out due to limited sample sizes. This work establishes a comparable framework and theoretical foundation for applying quantum energy models to time series prediction.

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.

Problem

• quantum advantage
• time-series forecasting
• conditional Boltzmann machines
• variational quantum algorithms
• hyperparameter optimisation

Innovation

• Variational Quantum Algorithm
• Conditional Boltzmann Machine
• Symmetric Hyperparameter Optimization
• Time-Series Forecasting
• Quantum-Classical Hybrid Architecture