  Generative models are a class of machine learning models that aim to learn
the underlying probability distribution of data. Unlike discriminative models,
generative models focus on capturing the data's inherent structure, allowing
them to generate new samples that resemble the original data. To fully exploit
the potential of modeling probability distributions using quantum physics, a
quantum-inspired generative model known as the Born machines have shown great
advancements in learning classical and quantum data over matrix product
state(MPS) framework. The Born machines support tractable log-likelihood,
autoregressive and mask sampling, and have shown outstanding performance in
various unsupervised learning tasks. However, much of the current research has
been centered on improving the expressive power of MPS, predominantly embedding
each token directly by a corresponding tensor index. In this study, we
generalize the embedding method into trainable quantum measurement operators
that can be simultaneously honed with MPS. Our study indicated that combined
with trainable embedding, Born machines can exhibit better performance and
learn deeper correlations from the dataset.
