Quantum Generative Adversarial Learning

Generative adversarial networks represent a powerful tool for classical machine learning: a generator tries to create statistics for data that mimics those of a true data set, while a discriminator tries to discriminate between the true and fake data. The learning process for generator and discriminator can be thought of as an adversarial game, and under reasonable assumptions, the game converges to the point where the generator generates the same statistics as the true data and the discriminator is unable to discriminate between the true and the generated data. This Letter introduces the notion of quantum generative adversarial networks, where the data consist either of quantum states or of classical data, and the generator and discriminator are equipped with quantum information processors. We show that the unique fixed point of the quantum adversarial game also occurs when the generator produces the same statistics as the data. Neither the generator nor the discriminator perform quantum tomography; linear programing drives them to the optimal. Since quantum systems are intrinsically probabilistic, the proof of the quantum case is different from—and simpler than—the classical case. We show that, when the data consist of samples of measurements made on high-dimensional spaces, quantum adversarial networks may exhibit an exponential advantage over classical adversarial networks.

Figure 1

Schematic of a general QGAN protocol. The ultimate goal of this adversarial game is for the discriminator ($D$) to determine whether the input data are real ($\sigma$) or fake ($\rho$). Here, $G$ is the generator which generates fake data hoping to fool the discriminator. We consider a variety of QGAN situations. First, where the real data are quantum, the generator is quantum (and hence, generates fake quantum data), and the discriminator is quantum. Second, the real data are quantum, the generator is classical, and the discriminator is either classical or quantum. Finally, we consider the case where the real data are purely classical and the generator and discriminator are both quantum.