Reduced-order modeling of two-dimensional turbulent Rayleigh-B'{e}nard 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'{e}nard convection flow at a Rayleigh number Ra = 105 and a Prandtl number Pr = 10. The 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. The 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.