Communications Physics

Expressive equivalence of classical and quantum restricted Boltzmann machines

Maria Demidik, Cenk Tüysüz, Nico Piatkowski, Michele Grossi, Karl Jansen

Published Oct 29, 2025• 0 citations

Abstract

Abstract The development of generative models for quantum machine learning has faced challenges such as trainability and scalability. A notable example is the quantum restricted Boltzmann machine (QRBM), where non-commuting Hamiltonians make gradient evaluation computationally demanding, even on fault-tolerant devices. In this work, we propose a semi-quantum restricted Boltzmann machine (sqRBM), a model designed to overcome difficulties associated with QRBMs. The sqRBM Hamiltonian commutes in the visible subspace while remaining non-commuting in the hidden subspace, enabling us to derive closed-form expressions for output probabilities and gradients. Our analysis shows that, for learning a given distribution, a classical model requires three times more hidden units than an sqRBM. Numerical simulations with up to 100 units validate this prediction. With reduced resource demands, sqRBMs provide a feasible framework for early quantum generative models.

DOI: 10.1038/s42005-025-02353-1

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