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We present a quantum algorithm that analyzes time series data simulated by aquantum differential equation solver. The proposed algorithm is a quantumversion of the dynamic mode decomposition algorithm used in diverse fields suchas fluid dynamics and epidemiology. Our quantum algorithm can also extractmatrix eigenvalues by analyzing the corresponding linear dynamical system. Ouralgorithm handles a broader range of matrices with complex eigenvalues, unlikeexisting efficient quantum eigensolvers limited to specific matrix types. Thecomplexity of our quantum algorithm is $O(\operatorname{poly}\log N)$ for an$N$-dimensional system. This is an exponential speedup over known classicalalgorithms with at least $O(N)$ complexity. Thus, our quantum algorithm isexpected to enable high-dimensional dynamical systems analysis and large matrixeigenvalue decomposition, intractable for classical computers.

Quantum Algorithm for Dynamic Mode Decomposition and Matrix Eigenvalue Decomposition with Complex Eigenvalues