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Two hybrid quantum-classical reservoir computing models are presented toreproduce low-order statistical properties of a two-dimensional turbulentRayleigh-B\'enard convection flow at a Rayleigh number Ra=1e+5 and a Prandtlnumber Pr=10. These properties comprise the mean vertical profiles of the rootmean square velocity and temperature and the turbulent convective heat flux.Both quantum algorithms differ by the arrangement of the circuit layers of thequantum reservoir, in particular the entanglement layers. The second of the twoquantum circuit architectures, denoted as H2, enables a complete execution ofthe reservoir update inside the quantum circuit without the usage of externalmemory. Their performance is compared with that of a classical reservoircomputing model. Therefore, all three models have to learn the nonlinear andchaotic dynamics of the turbulent flow at hand in a lower-dimensional latentdata space which is spanned by the time-dependent expansion coefficients of the16 most energetic Proper Orthogonal Decomposition (POD) modes. These trainingdata are generated by a POD snapshot analysis from direct numerical simulationsof the original turbulent flow. All reservoir computing models are operated inthe reconstruction mode. We analyse different measures of the reconstructionerror in dependence on the hyperparameters which are specific for the quantumcases or shared with the classical counterpart, such as the reservoir size andthe leaking rate. We show that both quantum algorithms are able to reconstructthe essential statistical properties of the turbulent convection flowsuccessfully with similar performance compared to the classical reservoirnetwork. Most importantly, the quantum reservoirs are by a factor of 4 to 8smaller in comparison to the classical case.

Reduced-order modeling of two-dimensional turbulent Rayleigh-B\'enard flow by hybrid quantum-classical reservoir computing