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We investigate the circuit complexity and Loschmidt echo for the (inverted) harmonic oscillators. Using a perturbative approach, we analytically derive the Lyapunov exponent and scrambling time of the inverted harmonic oscillators. We show that the circuit complexity and Loschmidt echo exhibit qualitatively similar behaviors, particularly the consistent Lyapunov exponent.
Qu et al. (Thu,) studied this question.