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The quantum circuit Born machine (QCBM) is a quantum physics inspiredimplicit generative model naturally suitable for learning binary images, with apotential advantage of modeling discrete distributions that are hard tosimulate classically. As data samples are generated quantum-mechanically, QCBMsencompass a unique optimization landscape. However, pioneering works on QCBMsdo not consider the practical scenario where only small batch sizes are allowedduring training. QCBMs trained with a statistical two-sample test objective inthe image space require large amounts of projective measurements to approximatethe model distribution well, unpractical for large-scale quantum systems due tothe exponential scaling of the probability space. QCBMs trained adversariallyagainst a deep neural network discriminator are proof-of-concept models thatface mode collapse. In this work we investigate practical learning of QCBMs. Weuse the information-theoretic \textit{Maximal Coding Rate Reduction} (MCR$^2$)metric as a second moment matching tool and study its effect on mode collapsein QCBMs. We compute the sampling based gradient of MCR$^2$ with respect toquantum circuit parameters with or without an explicit feature mapping. Weexperimentally show that matching up to the second moment alone is notsufficient for training the quantum generator, but when combined with the classprobability estimation loss, MCR$^2$ is able to resist mode collapse. Inaddition, we show that adversarially trained neural network kernel for infinitemoment matching is also effective against mode collapse. On the Bars andStripes dataset, our proposed techniques alleviate mode collapse to a largerdegree than previous QCBM training schemes, moving one step closer towardspracticality and scalability.

Sample-efficient Quantum Born Machine through Coding Rate Reduction