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Toward sampling from undirected probabilistic graphical models using a D-Wave quantum annealer

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Abstract

A D-Wave quantum annealer (QA) having a 2048 qubit lattice, with no missing qubits and couplings, allowed embedding of a complete graph of a restricted Boltzmann machine (RBM). A handwritten digit OptDigits dataset having 8 × 7 pixels of visible units was used to train the RBM using classical contrastive divergence. Embedding of the classically trained RBM into the D-Wave lattice was used to demonstrate that the QA offers a high-efficiency alternative to the classical Markov chain Monte Carlo (MCMC) for reconstructing missing labels of the test images as well as a generative model on a classically trained RBM. At any training iteration, the D-Wave-based classification had classification error less than half of that in MCMC. The main goal of this study was to investigate the quality of the QA sample from the RBM model probability distribution and compare it to a classical MCMC during the Gibbs sampling traditionally used in RBM training. For the OptDigits dataset, the states in the D-Wave sample belonged to about twice as many local valleys (LVs) as in the MCMC sample. All the lowest energy (the highest joint probability), local minima in the MCMC sample were also found by the D-Wave. The D-Wave missed many of the higher-energy LVs, while also finding many “new” LVs consistently missed by the MCMC. It was established that the “new” LVs that the D-Wave finds are important for the model distribution in terms of the energy of the corresponding local minima, the width of the LVs and the height of the escape barrier.

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Acknowledgements

The authors thank D-Wave Systems for access to their 2000 Q machine. This material is based on research sponsored by the Air Force Research Laboratory under agreement number FA8750-18-1-0096. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsement, either expressed or implied, of the Air Force Research Laboratory (AFRL) or the US Government.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, HPC2 Center for Computational Sciences, Mississippi State University, Mississippi State, MS, 39762, USA

    Yaroslav Koshka

  2. Department of Physics and Astronomy, HPC2 Center for Computational Sciences, Mississippi State University, Mississippi State, MS, 39762, USA

    M. A. Novotny

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  1. Yaroslav Koshka
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  2. M. A. Novotny
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Correspondence to Yaroslav Koshka.

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Koshka, Y., Novotny, M.A. Toward sampling from undirected probabilistic graphical models using a D-Wave quantum annealer. Quantum Inf Process 19, 353 (2020). https://doi.org/10.1007/s11128-020-02781-8

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