Quantum chaos and the complexity of spread of states
Vijay Balasubramanian, Pawel Caputa, Javier Magan, Qingyue Wu
TL;DR
The paper introduces spread complexity as a basis-agnostic measure of quantum state complexity, defined by minimizing the wavefunction's spread over all bases. It proves that a complete Krylov basis generated by the Lanczos recursion minimizes this cost (continuously near t=0 and discretely for time-evolved states) and connects the measure to an entropy-based interpretation. Computationally, it offers two practical routes: obtaining the Krylov basis from the Hessenberg form of the Hamiltonian or deriving Lanczos coefficients from the survival amplitude. Applying the framework to diverse models (harmonic/inverted oscillators, group manifolds, the Schwarzian theory, SYK, and random matrices), the authors find four dynamical regimes—ramp, peak, slope, plateau—in chaotic settings, with the slope reflecting spectral rigidity and differentiating random matrix ensembles via the spectral form factor.
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.