  Bayesian learning is ubiquitous for implementing classification and
regression tasks, however, it is accompanied by computationally intractable
limitations when the feature spaces become extremely large. Aiming to solve
this problem, we develop a quantum bayesian learning framework of the
restricted Boltzmann machine in the quantum-enhanced feature spaces. Our
framework provides the encoding phase to map the real data and Boltzmann weight
onto the quantum feature spaces and the training phase to learn an optimal
inference function. Specifically, the training phase provides a physical
quantity to measure the posterior distribution in quantum feature spaces, and
this measure is utilized to design the quantum maximum a posterior (QMAP)
algorithm and the quantum predictive distribution estimator (QPDE). It is shown
that both quantum algorithms achieve exponential speed-up over their classical
counterparts. Furthermore, it is interesting to note that our framework can
figure out the classical bayesian learning tasks, i.e. processing the classical
data and outputting corresponding classical labels. And a simulation, which is
performed on an open-source software framework for quantum computing,
illustrates that our algorithms show almost the same classification performance
compared to their classical counterparts. Noting that the proposed quantum
algorithms utilize the shallow circuit, our work is expected to be implemented
on the noisy intermediate-scale quantum (NISQ) devices, and is one of the
promising candidates to achieve quantum supremacy.
