Solving nonlinear differential equations with differentiable quantum circuits

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