Nonlinear dynamics as a ground-state solution on quantum computers

TL;DR

This paper introduces a variational quantum algorithm that encodes space and time to solve nonlinear differential equations, enabling the entire evolution to be obtained from a single ground-state computation on current quantum hardware.

Contribution

It presents a novel spacetime encoding method for VQAs, along with an adaptive multigrid strategy to address barren plateaus, demonstrated on nonlinear PDEs like Burgers' equation.

Findings

Successfully implemented on IBM Q and Quantinuum hardware.

Accurately reproduces solutions of nonlinear PDEs.

Provides a scalable approach for quantum simulation of nonlinear dynamics.

Abstract

For the solution of time-dependent nonlinear differential equations, we present variational quantum algorithms (VQAs) that encode both space and time in qubit registers. The spacetime encoding enables us to obtain the entire time evolution from a single ground-state computation. We describe a general procedure to construct efficient quantum circuits for the cost function evaluation required by VQAs. To mitigate the barren plateau problem during the optimization, we propose an adaptive multigrid strategy. The approach is illustrated for the nonlinear Burgers equation. We classically optimize quantum circuits to represent the desired ground-state solutions, run them on IBM Q System One and Quantinuum System Model H1, and demonstrate that current quantum computers are capable of accurately reproducing the exact results.