Differentiable Quantum Circuits for Solving Differential Equations and Generating Time Series

I will discuss a quantum algorithm to solve nonlinear differential equations which is suited to near term quantum devices. For this method a function is defined via expectation values of a parametrised quantum circuit which makes use of feature map encoding and variational ansatz. This parametrised circuit can be considered as a quantum neural network and can be differentiated to represent the function derivative analytically by using the parameter shift rule. The circuit is trained using a hybrid classical-quantum workflow and quantum machine learning techniques to provide a representation of the solution to a given differential equation. I will introduce different strategies for imposing boundary conditions. Furthermore, a Chebyshev feature map which offers high expressivity will also be introduced. I will show the results of simulating this method to solve various systems of differential equations cumulating in solving an instance of the Navier-Stoke equations for a convergent-divergent nozzle and applying similar strategies for achieving trainable generative modelling in the form of Quantum Quantile Mechanics.