Solving nonlinear differential equations with differentiable quantum circuits

TL;DR

This paper introduces a quantum algorithm that employs differentiable quantum circuits and a spectral method to solve nonlinear differential equations efficiently, avoiding finite difference errors and enabling high-dimensional problem handling.

Contribution

It presents a novel hybrid quantum-classical workflow using differentiable quantum circuits for solving nonlinear differential equations with a spectral approach.

Findings

Successfully simulated Navier-Stokes equations with quantum circuits.

Demonstrated high expressivity with Chebyshev quantum feature map.

Achieved accurate density, temperature, and velocity profiles in fluid flow simulation.

Abstract

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.