Quantum simulation of a noisy classical nonlinear dynamics

TL;DR

This paper introduces a quantum algorithm for efficiently simulating complex nonlinear stochastic dissipative systems, demonstrating potential quantum advantage in modeling phenomena like fluid dynamics.

Contribution

The authors develop the first rigorous quantum algorithm capable of simulating strongly nonlinear systems with polynomial runtime in system size and evolution time.

Findings

Algorithm approximates correlation functions with bounded error

Runtime scales polynomially with system parameters

Numerical experiments simulate 2D Navier-Stokes vortex flow

Abstract

We present an end-to-end quantum algorithm for simulating nonlinear dynamics described by a system of stochastic dissipative differential equations with a quadratic nonlinearity. The stochastic part of the system is modeled by a Gaussian noise in the equation of motion and in the initial conditions. Our algorithm can approximate the expected value of any correlation function that depends on $O(1)$ variables with rigorous bounds on the approximation error. The runtime scales polynomially with $\log{N}$, $t$, $J$, and $\lambda_1^{-1}$, where $N$ is the total number of variables, $t$ is the evolution time, $J$ is the nonlinearity strength, and $\lambda_1$ is the smallest dissipation rate. However, the runtime scales exponentially with a parameter quantifying inverse relative error in the initial conditions. To the best of our knowledge, this is the first rigorous quantum algorithm capable of simulating strongly nonlinear systems with $J\gg \lambda_1$ at the cost poly-logarithmic in $N$ and polynomial in $t$. The considered simulation problem is shown to be BQP-complete, providing a strong evidence for a quantum advantage. We benchmark the quantum algorithm via numerical experiments by simulating a vortex flow in the 2D Navier Stokes equation.