Quantum simulation of a noisy classical nonlinear dynamics

• URL: http://arxiv.org/abs/2507.06198v2
• Date: Fri, 03 Oct 2025 19:29:50 GMT
• Title: Quantum simulation of a noisy classical nonlinear dynamics
• Authors: Sergey Bravyi, Robert Manson-Sawko, Mykhaylo Zayats, Sergiy Zhuk,
• Abstract summary: We present an end-to-end quantum algorithm for simulating nonlinear dynamics described by a system of dissipative differential equations with a quadratic nonlinearity.<n>Our algorithm can approximate the expected value of any correlation function that depends on $O(1)$ variables with rigorous approximation error.<n>To the best of our knowledge, this is the first rigorous quantum algorithm capable of simulating strongly nonlinear systems with $Jgg lambda$ at the cost poly-logarithmic in $N$ and in $t$.
• Score: 2.0065923589074735
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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