Quantum algorithms for stochastic nonlinear differential equations

Abstract:Stochastic nonlinear dynamics underlie many models in engineering and computational physics, yet accurate high-dimensional simulation remains challenging. We present a quantum algorithm for a broad class of $N$-dimensional stochastic differential equations with dissipation and quadratic drift. The algorithm applies to strongly nonlinear systems with all-to-all interactions, thereby extending the scope of previously known quantum algorithms that were limited to weak nonlinearity and sparse systems. For norm-preserving drifts, a condition satisfied by key fluid dynamics discretizations, our method approximates expectation values of low-order correlation functions with rigorous error bounds at a cost polynomial in $\log{(N)}$ and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of $N$ interacting quantum harmonic oscillators with cost polylogarithmic in $N$. Finally, we formulate turbulence models, including Navier-Stokes and damped Euler equations, within this framework, opening a route to quantum simulation of strongly nonlinear SDEs governing turbulence and nonlinear wave dynamics.

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn)

arXiv:2606.08349 [quant-ph]

(or arXiv:2606.08349v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.08349 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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