Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems

TL;DR

This paper establishes fundamental limitations on quantum algorithms for solving nonlinear dynamical systems, especially chaotic ones, showing that such algorithms cannot efficiently simulate systems with positive Lyapunov exponents due to exponential complexity growth.

Contribution

It tightens the bounds of the quantum Carleman linearisation algorithm and proves that quantum algorithms cannot efficiently approximate solutions of chaotic systems with exponential complexity.

Findings

Quantum Carleman linearisation bounds are tightened.

Any quantum algorithm approximating solutions must have exponential complexity for chaotic systems.

Efficient quantum simulation of chaotic systems is likely impossible.

Abstract

We investigate the limitations of quantum computers for solving nonlinear dynamical systems. In particular, we tighten the worst-case bounds of the quantum Carleman linearisation (QCL) algorithm [Liu et al., PNAS 118, 2021] answering one of their open questions. We provide a further significant limitation for any quantum algorithm that aims to output a quantum state that approximates the normalized solution vector. Given a natural choice of coordinates for a dynamical system with one or more positive Lyapunov exponents and solutions that grow sub-exponentially, we prove that any such algorithm has complexity scaling at least exponentially in the integration time. As such, an efficient quantum algorithm for simulating chaotic systems or regimes is likely not possible.