Researchers at University of Exeter have proposed a quantum algorithm to solve systems of nonlinear differential equations.
Using a quantum feature map encoding, they have defined functions as expectation values of parametrized quantum circuits. They used 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 have described a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, they showed how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space.
From a technical perspective, they have designed a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity.
They simulated 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.
