A time-marching quantum algorithm for simulation of the nonlinear Lorenz dynamics

TL;DR

This paper introduces a quantum algorithm for simulating the nonlinear Lorenz system, enabling efficient time evolution with a recursive structure and preserving quantum speed-up, while accurately capturing chaotic and regular dynamics.

Contribution

The authors develop a novel quantum algorithm for second order discretized Lorenz dynamics that improves efficiency and maintains quantum advantages over previous methods.

Findings

The algorithm accurately reproduces Lorenz attractors and chaos.

It requires only a linear number of initial state copies relative to time steps.

Classical implementation confirms the algorithm's effectiveness.

Abstract

Simulating nonlinear classical dynamics on a quantum computer is an inherently challenging task due to the linear operator formulation of quantum mechanics. In this work, we provide a systematic approach to alleviate this difficulty by developing a quantum algorithm that implements the time evolution of a second order time-discretized version of the Lorenz model. The Lorenz model is a celebrated system of nonlinear ordinary differential equations that has been extensively studied in the contexts of climate science, fluid dynamics, and chaos theory. Our algorithm possesses a recursive structure and requires only a linear number of copies of the initial state with respect to the number of integration time-steps. This provides a significant improvement over previous approaches, while preserving the characteristic quantum speed-up in terms of the dimensionality of the underlying differential equations system, that similar time-marching quantum algorithms have previously demonstrated. Notably, by classically implementing the proposed algorithm, we showcase that it accurately captures the structural characteristics of the Lorenz system, reproducing both regular attractors--limit cycles--and the chaotic attractor within the chosen parameter regime.