  We propose a quantum algorithm to solve systems of nonlinear differential
equations. Using a quantum feature map encoding, we define functions as
expectation values of parametrized quantum circuits. We use automatic
differentiation to represent function derivatives in an analytical form as
differentiable quantum circuits (DQCs), thus avoiding inaccurate finite
difference procedures for calculating gradients. We describe a hybrid
quantum-classical workflow where DQCs are trained to satisfy differential
equations and specified boundary conditions. As a particular example setting,
we show how this approach can implement a spectral method for solving
differential equations in a high-dimensional feature space. From a technical
perspective, we design a Chebyshev quantum feature map that offers a powerful
basis set of fitting polynomials and possesses rich expressivity. We simulate
the algorithm to solve an instance of Navier-Stokes equations, and compute
density, temperature and velocity profiles for the fluid flow in a
convergent-divergent nozzle.
