Chaos and Complexity in Quantum Mechanics

TL;DR

This work introduces a circuit-complexity based probe of quantum chaos, leveraging a covariance-matrix formalism for Gaussian states to study a forward-backward evolution protocol with slightly perturbed Hamiltonians. Using a tunable oscillator model that transitions from regular to inverted (chaotic) dynamics, the authors show that complexity exhibits a delayed onset followed by linear growth in the chaotic regime, with the slope and pickup time mapping onto the Lyapunov exponent and scrambling time, respectively, in agreement with OTOC analyses. They extend the analysis to a field-theory setting via lattice discretization, demonstrating that the complexity diagnostic can capture scrambling in many-body systems. The results offer a complementary, information-theoretic perspective on quantum chaos with potential ties to holographic complexity notions and future applications to strongly coupled quantum systems.

Abstract

We propose a new diagnostic for quantum chaos. We show that time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behaviour.

Figures (8)

• Figure 1: $\mathcal{\hat{C}}(\tilde{U})$ vs time for Regular Oscillator ($m=1,\lambda=1.2,\delta\lambda=0.01$)
• Figure 2: $\mathcal{\hat{C}}(\tilde{U})$ vs time for Inverted Oscillator ($m =1, \lambda= 15, \delta\lambda =0.01$)
• Figure 3: $\mathcal{\hat{C}}(\tilde{U})$ vs time for different values of $\lambda$ (with $\delta\lambda=0.01,m=1$)
• Figure 4: Complexity vs $\lambda$ for different time. (a) In the left figure, for all $t$, $\hat{\cal C}(\tilde{U})$ starts to increase before the critical of $\lambda$ namely $\lambda=m^2.$ (b) In the right figure, we can observe that near $t=40$ there is a sharp increase in $\hat{\cal C}(\tilde{U})$ at the critical value of $\lambda$ ($\lambda=1$, for the choice of the parameter). We have set $\delta \lambda=0.01$ and $m=1$ for both the figures.
• Figure 5: (a) Slope $\phi$ vs $\lambda$ ($\delta=0.01,m=1$), (b)$t_s$ vs $\lambda$($\delta=0.01,m=1$)