  We present a quantum algorithm that analyzes time series data simulated by a
quantum differential equation solver. The proposed algorithm is a quantum
version of the dynamic mode decomposition algorithm used in diverse fields such
as fluid dynamics and epidemiology. Our quantum algorithm can also compute
matrix eigenvalues and eigenvectors by analyzing the corresponding linear
dynamical system. Our algorithm handles a broad range of matrices, in
particular those with complex eigenvalues. The complexity of our quantum
algorithm is $O(\operatorname{poly}\log N)$ for an $N$-dimensional system. This
is an exponential speedup over known classical algorithms with at least $O(N)$
complexity. Thus, our quantum algorithm is expected to enable high-dimensional
dynamical systems analysis and large matrix eigenvalue decomposition,
intractable for classical computers.
