Empirical investigation of the low temperature energy function of the Restricted Boltzmann Machine using a 1000 qubit D-Wave 2X

Abstract

D-Wave 2X with more than 1000 qubits was applied to the relatively rugged energy landscape of trained Restricted Boltzmann Machines (RBMs). The D-Wave machine has a Chimera interconnect architecture. A native RBM restricted to the Chimera graph was found difficult to train for large number of RBM units. To overcome this difficulty, a RBM embedding that combined qubits in order to significantly increase the connectivity between hidden and visible units of the RBM was investigated. The results for the lowest-energy and some of the higher-energy states found by D-Wave 2X were compared with those of the classical simulated annealing (SA) algorithm. In many cases, the D-Wave machine successfully found the same RBM lowest energy state as that found by SA. In the relatively simple cases of training patterns investigated in this work, the lowest energy state also corresponded to one of the training patterns. Sometimes the D-Wave machine returned a state corresponding to one of the higher energy states found by SA. The inherently non-perfect embedding of the RBM into the Chimera lattice used in this work (i.e., multiple qubits combined into a single RBM unit were not guaranteed to have perfectly aligned qubits) and the existence of small bias fields in the D-Wave hardware were found to be responsible for the discrepancy in the D-Wave and the SA results. In some but not all of the investigated cases, introduction of a small bias field into the energy function or optimization of the chain-strength parameter in the D-Wave embedding successfully addressed difficulties of the particular RBM embedding.