Quantum chaos and the complexity of spread of states

Abstract

We propose a measure of quantum state complexity defined by minimizing the spread of the wavefunction 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.

Krylov complexity from integrability to chaos

“…In addition to the behavior of K-complexity for given individual quantum systems, such as the SYK model [3,5,6], 2D CFTs [17,18], and more general symmetry-based Hamiltonian systems [19,20], it is interesting and important to categorize the possible Krylov phenomenologies according to more universal criteria. One of the most interesting of these is clearly the behavior of K-complexity in the class of chaotic quantum systems as opposed to that of integrable ones, initiated in [7] for systems away from the thermodynamic limit.…”

“…A mathematically precise definition of the complexity of time evolution under a given Hamiltonian is given by Krylov complexity [3][4][5] or 'K-complexity' for short. To date, several aspects of Krylov complexity have been studied in various setups and systems, for example [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Krylov complexity in saddle-dominated scrambling

“…Interestingly, the validity of this hypothesis goes beyond the semi-classical regime, where OTOC or, more specifically, the Lyapunov exponent is ill-defined. In recent years, the study of operator growth and K-complexity has received significant attention from many-body systems to the conformal field theories and black hole physics [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55].…”

Krylov complexity in saddle-dominated scrambling

“…Interestingly, the validity of this hypothesis goes beyond the semi-classical regime, where OTOC or, more specifically, the Lyapunov exponent is ill-defined. In recent years, the study of operator growth and K-complexity has received significant attention from many-body systems to the conformal field theories and black hole physics [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. The growth of Lanczos coefficients essentially captures the time evolution of the Kcomplexity (see figure 1).…”