Quantum chaos and the complexity of spread of states

We propose a measure of quantum state complexity defined by minimizing the spread of the wave function over all choices of basis. Our measure is controlled by the survival amplitude for a state to remain unchanged, and can be efficiently computed in theories with discrete spectra. For continuous Hamiltonian evolution, it generalizes Krylov operator complexity to quantum states. We apply our methods to the harmonic and inverted oscillators, particles on group manifolds, the Schwarzian theory, the SYK model, and random matrix models. For time-evolved thermofield double states in chaotic systems our measure shows four regimes: a linear ramp up to a peak that is exponential in the entropy, followed by a slope down to a plateau. These regimes arise in the same physics producing the slope-dip-ramp-plateau structure of the spectral form factor. Specifically, the complexity slope arises from spectral rigidity, distinguishing different random matrix ensembles.

Automated summary

We propose a measure of quantum state complexity by minimizing the spread of wave function over different choices of basis. This measure is controlled by the survival amplitude for a state to remain unchanged and can be efficiently computed in theories with discrete spectra. It generalizes Krylov operator complexity to quantum states for continuous Hamiltonian evolution. By applying our method to various systems, we reveal four regimes in the time-evolved thermofield double states, showing the same physics as the spectral form factor's slope-dip-ramp-plateau structure.