Learning to generate high-dimensional distributions with low-dimensional quantum Boltzmann machines

arXiv (Cornell University)

Abstract

In recent years, researchers have been exploring ways to generalize Boltzmann machines (BMs) to quantum systems, leading to the development of variations such as fully-visible and restricted quantum Boltzmann machines (QBMs). Due to the non-commuting nature of their Hamiltonians, restricted QBMs face trainability issues, whereas fully-visible QBMs have emerged as a more tractable option, as recent results demonstrate their sample-efficient trainability. These results position fully-visible QBMs as a favorable choice, offering potential improvements over fully-visible BMs without suffering from the trainability issues associated with restricted QBMs. In this work, we show that low-dimensional, fully-visible QBMs can learn to generate distributions typically associated with higher-dimensional systems. We validate our findings through numerical experiments on both artificial datasets and real-world examples from the high energy physics problem of jet event generation. We find that non-commuting terms and Hamiltonian connectivity improve the learning capabilities of QBMs, providing flexible resources suitable for various hardware architectures. Furthermore, we provide strategies and future directions to maximize the learning capacity of fully-visible QBMs.

Authors 8

1. Cenk Tüysüz
2. Maria Demidik

   Cyprus Institute · Deutsches Elektronen-Synchrotron DESY

   Affiliation as printed

   CQTA , Deutsches Elektronen-Synchrotron DESY , Platanenallee 6 , 15738 Zeuthen , Germany

   Computation-Based Science and Technology Research Center , The Cyprus Institute , 20 Kavafi Street , 2121 Nicosia , Cyprus

3. Luuk Coopmans
   Affiliation as printed

   Quantinuum , Partnership House , Carlisle Place , London SW1P 1BX , United Kingdom

4. Enrico Rinaldi

   RIKEN · RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences

   Affiliation as printed

   Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) Program , RIKEN , Wako , Saitama 351-0198 , Japan

   Quantinuum K.K. , Otemachi Financial City Grand Cube 3F , 1-9-2 Otemachi , Chiyoda-ku , Tokyo , Japan

5. V. Croft Aachen Applied Quantum Algorithms Leiden aQa L and Leiden Institute of Advanced Computer Science (LIACS)

   Leiden University

   Affiliation as printed

   Applied Quantum Algorithms Leiden aQa L and Leiden Institute of Advanced Computer Science (LIACS) , Leiden University , Niels Bohrweg 2 , 2333 CA Leiden , Netherlands

6. Y. Haddad

   Northeastern University

   Affiliation as printed

   Department of Physics , Northeastern University , Boston , MA 02115 USA

7. Matthias Rosenkranz
   Affiliation as printed

   Quantinuum , Partnership House , Carlisle Place , London SW1P 1BX , United Kingdom

8. Karl Jansen

   Cyprus Institute · Deutsches Elektronen-Synchrotron DESY

   Affiliation as printed

   CQTA , Deutsches Elektronen-Synchrotron DESY , Platanenallee 6 , 15738 Zeuthen , Germany

   Computation-Based Science and Technology Research Center , The Cyprus Institute , 20 Kavafi Street , 2121 Nicosia , Cyprus