Solving nonlinear differential equations with differentiable quantum circuits

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.

Automated summary

A quantum algorithm is proposed to solve systems of nonlinear differential equations using a quantum feature map encoding and differentiable quantum circuits. The hybrid quantum-classical workflow involves training DQCs to meet differential equations and specified boundary conditions, with a particular example demonstrating a spectral method in a high-dimensional feature space. The use of a Chebyshev quantum feature map offers a powerful basis set for fitting polynomials with rich expressivity, as shown in a simulation solving Navier-Stokes equations for fluid flow.