Quantum algorithms for stochastic nonlinear differential equations

• URL: http://arxiv.org/abs/2606.08349v1
• Date: Sat, 06 Jun 2026 21:45:53 GMT
• Title: Quantum algorithms for stochastic nonlinear differential equations
• Abstract summary: We present a quantum subroutine for a broad class of $N$-dimensional differential equations with dissipation and quadratic drift.<n>The norm-preserving drifts applies to strongly nonlinear systems with all-to-all interactions.<n>Our method approximates expectation values of low-order correlation functions with rigorous error bounds.
• Score: 2.0065923589074735
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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