Quantum Algorithms for Stochastic Nonlinear Differential Equations

We consider the problem of simulating dynamics of classical nonlinear dissipative systems with N degrees of freedom. To make the problem tractable for quantum computers, we add a weak Gaussian noise to the equation of motion and to the initial state. Our main result is an end-to-end quantum algorithm for simulating the noisy dynamics of nonlinear systems satisfying certain sparsity and divergence-free conditions. For any constant nonzero noise rate, the quantum runtime scales polynomially with log(N), evolution time, inverse error tolerance, and the relative strength of nonlinearity and dissipation. Our main technical tool is the Kolmogorov partial differential equation describing time evolution of scalar functions of solutions averaged over noise.

Joint work with Robert Manson-Sawko, Mykhaylo Zayats, and Sergiy Zhuk

Preprint: arXiv:2507.06198