# Objective function of Quantum GAN in the paper "Quantum generative adversarial networks"

In this paper about Quantum GANs, the authors do not explain clearly how do they have the equation

$$\newcommand{\tr}{\operatorname{Tr}}\newcommand{\Pr}{\operatorname{Pr}} V(\vec{\theta}_D, \vec{\theta}_G)= \frac{1}{2}+\frac{1}{2\Lambda} \sum_{\lambda=1}^{\Lambda} \left[ \cos^2(\phi)\tr(Z\rho_\lambda^{DR}(\vec{\theta_R}))-\sin^2(\phi)\tr(Z\rho_\lambda^{DG}(\vec{\theta_R},\vec{\theta_D,z})) \right],$$

from

$$V(\vec{\theta}_D, \vec{\theta}_D)= \frac{1}{\Lambda} \sum_{\lambda=1}^{\Lambda} \Pr\Big[ ( D(\vec{\theta}_D,|\lambda\rangle, R(|\lambda\rangle)) = |\operatorname{real}\rangle ) \\\cap ( D(\vec{\theta}_D,|\lambda\rangle, G(\vec{\theta}_G,|\lambda,z\rangle)) =|\operatorname{fake}\rangle ) \Big] .$$

They suggested that it relate to classical case, but I still don't know the key point.

Hope to see some ideas from you. Thanks!

In essence, we want to maximize the probability that the discriminator $$D$$ outputs the state $$\left|\textrm{Real}\right\rangle$$ on training data $$R$$ and $$\left|\textrm{Fake}\right\rangle$$ when given a state from the generator $$G$$. This roughly matches the interpretation of the optimization problem over the constructed $$\left\langle Z\right\rangle$$ expectation value. In hindsight, it would have made more sense to use the union operator $$\cup$$ in Eq. 3 instead of the intersection operator $$\cap$$. It might also have been worth to comment that we did not use log probabilities (which would be interesting to do along with an analysis of the sampling complexity).
Additionally, the “$$\phi$$” terms of Eq. 11 may be somewhat confusing. This relative frequency is a free parameter in the construction. I presumed the cost function should be normalized over the relative frequency using the training data $$R$$ and using the generator $$G$$. This is an artefact of an earlier derivation where $$R$$ and $$G$$ were inputs on different channels of $$D$$ and the which-path information had to be erased. Leaving it in the single channel formulation has the advantage of preventing pathological cases where the cost function would only be measured on $$R$$ (or $$G$$). As told in the main text, the simple case where $$R$$ and $$G$$ are picked at random corresponds to $$\phi = \frac{\pi}{4}$$.