Natural Gradient Optimization for Optical Quantum Circuits

Optical quantum circuits can be optimized using gradient descent methods, as the gates in a circuit can be parametrized by continuous parameters. However, the parameter space as seen by the cost function is not Euclidean, which means that the Euclidean gradient does not generally point in the direction of steepest ascent. In order to retrieve the steepest ascent direction, it is possible to take into account the local metric in the space parameters in what is known as the Natural Gradient (NG) method. In this work we implement the Natural Gradient (NG) for optical quantum circuits. In particular, we adapt the NG approach to a complex-valued parameter space. We then compare the NG approach to vanilla gradient descent and to Adam over two state preparation tasks: a singlephoton source and a Gottesman-Kitaev-Preskill state source. We observe that the NG approach converges faster (due in part to the possibility of using larger learning rates) and with a significantly smoother decay of the cost function throughout the optimization.