Statistics of Classical Nonlinear Dynamical Systems Obtained from Noisy Intermediate-Scale Quantum Devices

Classical nonlinear dynamical systems are often characterized by their steady state probability distribution functions (PDFs). The PDF can be obtained by the accumulation of statistics, or more directly as a solution to the linear Fokker-Planck Equation (FPE). However, except in special cases, the FPE cannot be solved analytically, thus necessitating computational approaches. The increasing availability of NISQ devices motivates us to investigate their utility in this context. We use the Quantum Phase Estimation algorithm to obtain the stationary solution of the FPE, apart from information about the sign of the amplitudes. We implement the algorithm on an 11-qubit device built by IonQ and test the case of the one-dimensional nonlinear Ornstein-Uhlenbeck systems. Results from quantum computing are presented along with the methods used for error correction, highlighting both the possibility of solving the FPE on quantum devices and challenges that remain.