A bstract Heisenberg time evolution under a chaotic many-body Hamiltonian H transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or ‘K-complexity’, quantifies this growth with respect to a special basis, generated by H by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time t s > log( S ). We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK 4 model, which is maximally chaotic, and compare the results with the SYK 2 model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.
Recently, the propagation of information through quantum many-body systems, developed to study quantum chaos, have found many applications from black holes to disordered spin systems. Among other quantitative tools, Krylov complexity has been explored as a diagnostic tool for information scrambling in quantum many-body systems. We introduce a universal limit to the growth of the Krylov complexity in dissipative open quantum systems by utilizing the uncertainty relation for non-Hermitian operators. We also present the analytical results of Krylov complexity for characteristic behavior of Lanczos coefficients in dissipative systems. The validity of these results is demonstrated by explicit study of transverse-field Ising model under dissipative effects.
Abstract Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.
Krylov complexity, as a novel measure of operator complexity under Heisenberg evolution, exhibits many interesting universal behaviors and also bounds many other complexity measures. In this work, we study Krylov complexity
K(t) in Random Matrix Theory (RMT). In large
N limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient
{bn} saturate to constant plateau
n→∞limbn=b, rendering a linear growing complexity
K(t)∼t, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value
b to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the
{bn} in chaotic local quantum systems. Therefore we conjecture that in chaotic local quantum systems after scrambling time, the speed of linear growth of Krylov complexity cannot be larger than that in RMT. (2) For low temperature, we analytically show that
bn will first exhibit linear growth with
n, whose slope saturates the famous chaos bound. After hitting the same plateau
b,
bn will then remain constant. This indicates
K(t)∼e2πt/β before scrambling time
t∗∼O(βlogβ), and after that it will grow linearly in time, with the same speed as in infinite temperature. We finally remark on the effect of finite
N corrections.
We study quantum dynamics generated by time-dependent Hamiltonians in Krylov space, the minimal subspace in which the evolution takes place. We establish a direct link between dynamics in the time-dependent Krylov subspace and the underlying Lie-algebraic structure of the Hamiltonian. We develop a general framework in which the dynamics in the time-dependent Krylov subspace is generated by ladder operators of the associated Lie algebra. In particular, we identify the minimal conditions under which the exact time-dependent Krylov dynamics is naturally determined by the interaction-picture Hamiltonian and governed by an embedded
sl(2,C) subalgebra. We further show that an exact single-exponential representation of the time-evolution operator gives rise to a distinct time-independent Krylov dynamics in a unitarily related basis, from which the exact time-dependent Krylov dynamics can nevertheless be recovered. We also extend the framework to the oscillator algebra as the simplest extension of the nilpotent Heisenberg--Weyl algebra, and provide further examples, including the translated and dilated harmonic oscillator, systems governed by closed Virasoro subalgebras, a spin in a rotating magnetic field, and higher-dimensional generalizations for multi-level systems. In addition, we introduce a new quantum speed limit to the complexity growth rate generated by a time-dependent generator and show that, for evolutions governed by a Lie algebra, it retains the same functional form as in the time-independent case. Remarkably, saturation of this bound is strongly affected by temporal driving and persists only when the Hamiltonian commutes with itself at different times. These results establish a unified framework for characterizing operator growth and Krylov complexity in time-dependent quantum systems with underlying Lie-algebraic structures.
Krylov methods have reappeared recently, connecting physically sensible notions of complexity with quantum chaos and quantum gravity. In these developments, the Hamiltonian and the Liouvillian are tridiagonalized so that Schrodinger/Heisenberg time evolution is expressed in the Krylov basis. In the context of Schrodinger evolution, this tridiagonalization has been carried out in Random Matrix Theory. We extend these developments to Heisenberg time evolution, describing how the Liouvillian can be tridiagonalized as well until the end of Krylov space. We numerically verify the analytical formulas both for Gaussian and non-Gaussian matrix models.