Chaos and complexity for inverted harmonic oscillators

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.

Automated summary

We investigate the circuit complexity and Loschmidt echo for the (inverted) harmonic oscillators using a perturbative approach. Analytical results for the Lyapunov exponent and scrambling time of the inverted harmonic oscillators are derived. Our findings show that the circuit complexity and Loschmidt echo exhibit qualitatively similar behaviors, particularly with respect to the consistent Lyapunov exponent.