  Quantum circuit Born machines are generative models which represent the
probability distribution of classical dataset as quantum pure states.
Computational complexity considerations of the quantum sampling problem suggest
that the quantum circuits exhibit stronger expressibility compared to classical
neural networks. One can efficiently draw samples from the quantum circuits via
projective measurements on qubits. However, similar to the leading implicit
generative models in deep learning, such as the generative adversarial
networks, the quantum circuits cannot provide the likelihood of the generated
samples, which poses a challenge to the training. We devise an efficient
gradient-based learning algorithm for the quantum circuit Born machine by
minimizing the kerneled maximum mean discrepancy loss. We simulated generative
modeling of the Bars-and-Stripes dataset and Gaussian mixture distributions
using deep quantum circuits. Our experiments show the importance of circuit
depth and gradient-based optimization algorithm. The proposed learning
algorithm is runnable on near-term quantum device and can exhibit quantum
advantages for generative modeling.
