Quantum chaos and the complexity of spread of states

TL;DR

This paper introduces a new measure of quantum state complexity based on wave-function spread minimization, applicable to various quantum systems, revealing distinct regimes in chaotic dynamics linked to spectral properties.

Contribution

It proposes a novel complexity measure for quantum states that generalizes Krylov operator complexity and applies it to multiple models, uncovering regimes in chaotic evolution.

Findings

Complexity exhibits four regimes: ramp, peak, slope, plateau.

Complexity slope is related to spectral rigidity and ensemble differences.

Method efficiently computes complexity in systems with discrete spectra.

Abstract

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.