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

Two hybrid quantum-classical reservoir computing models are presented to reproduce low-order statistical properties of a two-dimensional turbulent Rayleigh-B\'enard convection flow at a Rayleigh number Ra=1e5 and a Prandtl number Pr=10. Both quantum algorithms differ by the arrangement of the circuit layers in the quantum reservoir, in particular the entanglement layers. The second of the two architectures, denoted as H2, enables a complete execution of the reservoir update inside the quantum circuit. Their performance is compared with that of a classical reservoir computing model. All three models have to learn the nonlinear and chaotic dynamics of the flow in a lower-dimensional latent data space which is spanned by the time series of the 16 most energetic Proper Orthogonal Decomposition (POD) modes. These training data are generated by a POD snapshot analysis from the turbulent flow. All reservoir computing models are operated in the reconstruction or open-loop mode, i.e., they receive 3 POD modes as an input at each step and reconstruct the missing 13 ones. We analyse the reconstruction error in dependence on the hyperparameters which are specific for the quantum cases or shared with the classical counterpart, such as the reservoir size and the leaking rate. We show that both quantum algorithms are able to reconstruct essential statistical properties of the turbulent convection flow successfully with a small number of qubits of n<=9. These properties comprise the velocity and temperature fluctuation profiles and, in particular, the turbulent convective heat flux, which quantifies the turbulent heat transfer across the layer and manifests in coherent hot rising and cold falling thermal plumes.