Carleman Linearization Method Shows Promise for Quantum Simulation of Low-Reynolds Fluid Flows
Researchers have shown that the lowest (second) order truncation of Carleman linearization of fluid equations can recover the steady-state solution of fluid flows, though with decreasing accuracy at higher Reynolds numbers. The result was first proved analytically for a forced logistic equation and then validated numerically for two-dimensional Kolmogorov-like flows at Reynolds numbers below approximately 10. The finding may open pathways for quantum simulation of low-Reynolds steady-state fluid dynamics.
A new preprint posted to arXiv demonstrates that the second-order Carleman linearization (C2) of the Navier-Stokes-type fluid equations is sufficient to capture the long-term, steady-state behavior of fluid flows in the low Reynolds number regime. The authors first establish this property analytically using a decaying logistic equation with external forcing as a tractable model, then extend the validation to the more physically complex case of two-dimensional Kolmogorov-like flow. Numerical results confirm the asymptotic accuracy of C2 for Reynolds numbers below roughly Re ~ 10, while accuracy degrades as Reynolds number increases beyond this range. The significance of the result lies in the fact that Carleman linearization converts nonlinear differential equations into an infinite-dimensional linear system, and truncating at second order yields a form amenable to quantum linear algebra algorithms. The authors suggest this time-asymptotic property could make quantum computers a practical tool for simulating steady-state low-Reynolds fluid flows, a regime relevant to microfluidics and other applications.
What's missing
The study is a preprint and has not yet undergone peer review. The analysis is restricted to Reynolds numbers below ~10 and two-dimensional flow geometries; applicability to three-dimensional or higher-Reynolds flows remains unaddressed. The paper does not provide concrete resource estimates (qubit counts, circuit depths) for the proposed quantum simulation approach, leaving the practical feasibility of quantum implementation an open question.
What different sources said
- arXiv physicsCenter
Lowest order Carleman linearization for low Reynolds long-term behaviour of fluid flow simulations
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