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arXiv:2604.16179

Quantum-Inspired Simulation of 2D Turbulent Rayleigh-Bénard Convection

Nis-Luca van Hülst, Mario Guillaume Cecile, Hai-Yen Van, Tomohiro Hashizume, Eugene de Villiers, Dieter Jaksch

physics.flu-dynphysics.comp-phquant-phSubmitted Apr 17, 2026• Announced Apr 20, 2026• 0 citations
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Abstract

Turbulent thermal convection governs heat transport in systems ranging from stellar interiors to industrial heat exchangers. Two-dimensional Rayleigh-Bénard convection serves as a paradigm for these flows, reproducing key features such as thin boundary layers, large-scale circulation, and sustained plume dynamics. While Matrix Product State (MPS) methods have demonstrated significant compression of isothermal turbulent fields, their application to buoyancy-driven flows with active thermal coupling has remained unexplored. We apply MPS to two-dimensional Rayleigh-Bénard convection with dynamical simulations up to $\mathrm{Ra} = 10^{10}$. An a priori decomposition of DNS snapshots up to $\mathrm{Ra} = 10^{11}$ shows that the bond dimension $χ$ required to represent the flow fields grows without saturation, in contrast to the plateauing of $χ$ reported for velocity fields in isothermal 2D turbulence. Crucially, however, dynamical simulations solving the governing equations directly in the compressed MPS format at fixed $χ$ show that the $χ$ required to recover statistical observables, such as the Nusselt number, scales significantly more favorably with $\mathrm{Ra}$ than the a priori complexity suggests. At $\mathrm{Ra} = 10^{10}$, a relative error of $1.8\%$ in the mean Nusselt number is achieved with a nearly 9-fold reduction in degrees of freedom, using a $χ$ comparable to that required at $\mathrm{Ra} = 10^{9}$. Spectral analysis confirms the progressive recovery of spatial and temporal scales with increasing $χ$. These findings establish MPS as a scalable tool for simulating thermally driven turbulence, suggesting the method may remain viable for investigations of the ultimate regime at substantially higher $\mathrm{Ra}$.

Comments
10 figures, 14 pages

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Quantum turbulence spans length scales from the system size L to the healing length ξ , making direct numerical simulations (DNS) of the Gross-Pitaevskii (GP) equation computationally expensive when L ≫ ξ . We present a matrix-product state (MPS) solver for the GP equation that efficiently compresses the wavefunction by truncating weak interlength-scale correlations. This approach reduces memory usage by factors ranging from 10 × to over 10 000 × compared to DNS. We benchmark our approach on nonlinear excitations, namely dark solitons (1D) and quantized vortices (2D, 3D), capturing key dynamics such as Kelvin-wave propagation and vortex-ring emission in the case of vortex-line reconnection. For turbulent states composed of multiple nonlinear excitations, we find that the memory compression of the MPS representation is directly proportional to the soliton or vortex densities. We also accurately reproduce established results from two-point correlation functions and energy spectra, where we recover the incompressible kinetic energy spectrum with little memory overhead. These results demonstrate the representative capabilities of the MPS ansatz for quantum turbulence and pave the way for studying this nonequilibrium state using previously prohibited system sizes. Moreover, the MPS ansatz can be adapted to existing quantum algorithms, providing a framework to implementing our time-evolution method on near-future quantum processing units.