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A quantum-inspired approach to exploit turbulence structures

A preprint version of the article is available at arXiv.


Understanding turbulence is key to our comprehension of many natural and technological flow processes. At the heart of this phenomenon lies its intricate multiscale nature, describing the coupling between different-sized eddies in space and time. Here we analyze the structure of turbulent flows by quantifying correlations between different length scales using methods inspired from quantum many-body physics. We present the results for interscale correlations of two paradigmatic flow examples, and use these insights along with tensor network theory to design a structure-resolving algorithm for simulating turbulent flows. With this algorithm, we find that the incompressible Navier–Stokes equations can be accurately solved even when reducing the number of parameters required to represent the velocity field by more than one order of magnitude compared to direct numerical simulation. Our quantum-inspired approach provides a pathway towards conducting computational fluid dynamics on quantum computers.

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Fig. 1: Interscale correlations of turbulent fluid flows.
Fig. 2: 2D temporally developing jet.
Fig. 3: 3D Taylor–Green vortex.

Data availability

Our Code Ocean capsule49 contains the raw output data from our MPS simulations. These data were generated using the C functions tntMpsBoxTurbulence2DTimeEvolutionRK2(...) and tntMpsBoxTurbulence3DTimeEvolutionRK2(...), using the initial conditions and parameters defined in the Set-up of numerical experiments section in the Methods. Source data for Figs. 1, 2 and 3 are available via Code Ocean49.

Code availability

The MATLAB code required to reproduce Figs. 1, 2 and 3 is available via Code Ocean49. The MPS simulations were done using the Tensor Network Theory Library50.


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The work at the University of Oxford was supported by the EPSRC Programme Grant DesOEQ (EP/P009565/1), and M.K. and D.J. acknowledge financial support from the National Research Foundation, Prime Ministers Office, Singapore, and the Ministry of Education, Singapore, under the Research Centres of Excellence programme. D.J. also acknowledges support by the Excellence Cluster ‘The Hamburg Centre for Ultrafast Imaging—Structure, Dynamics and Control of Matter at the Atomic Scale’ of the Deutsche Forschungsgemeinschaft. Current work at the University of Pittsburgh is supported by the National Science Foundation under grant no. CBET-2042918. S.D. is thankful for support from the EPSRC New Investigator Award (EP/T031255/1) and New Horizons grant (EP/V04771X/1). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. We also acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work ( Finally, we thank the scientists and engineers at BAE Systems, Bristol for fruitful discussions and advice.

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Authors and Affiliations



D.J. conceived the research project and N.G., M.L., P.G. and D.J. jointly planned it. N.G., M.L., S.D., M.K. and D.J. developed the quantum-inspired measure for interscale correlations based on Schmidt decompositions and hierarchical lattices. N.G., M.L. and S.D. formulated the matrix product state algorithm and carried out the analytical calculations. N.G., Q.Y.v.d.B. and M.K. wrote the software. N.G., H.B. and P.G. designed the numerical experiments for comparing MPS, URDNS and DNS. N.G. performed the numerical experiments. N.G., M.L., S.D., H.B., P.G., M.K. and D.J. analyzed and interpreted the numerical results. N.G., M.K. and D.J. wrote the manuscript with contributions from M.L., S.D., H.B. and P.G., and Q.Y.v.d.B. helped revise the manuscript. The Supplementary Information was written by N.G., M.L., S.D. and M.K.. The project was supervised by D.J.

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Correspondence to Nikita Gourianov or Dieter Jaksch.

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Nature Computational Science thanks Koji Fukagata, Nuno Loureiro and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Editor recognition statement Handling editor: Jie Pan, in collaboration with the Nature Computational Science team.

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Supplementary Figs. 1–7 and sections 1–6.

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Gourianov, N., Lubasch, M., Dolgov, S. et al. A quantum-inspired approach to exploit turbulence structures. Nat Comput Sci 2, 30–37 (2022).

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