In the talk, I will discuss advances in the field of quantum scientific computing. This is based on a series of works for solving differential equations for applications in fluid dynamics and generative modelling. First, I will show how to differentiate quantum circuits with feature maps and embed derivatives of multidimensional functions. This approach relies on automatic differentiation to represent derivatives in an analytical form, thus avoiding inaccurate finite difference procedures. We refer to the underlying circuits as derivative quantum circuits (DQCs). I will describe the proposed hybrid quantum-classical workflow where DQCs are trained to satisfy nonlinear differential equations and specified boundary conditions.

As an example, I will apply the algorithm to solve a problem from computational fluid dynamics. This corresponds to the fluid flow in a convergent-divergent nozzle (described by Navier-Stokes equations), where we predict density, temperature and velocity profiles for the stiff system of equations that can challenge classical solvers. Second, I will also show that DQCs can provide an advantage for generative modelling from stochastic differential equations, giving access to sampling from financially relevant models. Finally, I will discuss the protocols that may be scaled to treat large industrial problems in the future.