Krylov complexity, a quantum complexity measure which uniquely characterizes the spread of a quantum state or an operator, has recently been studied in the context of quantum chaos. However, the definitiveness of this measure as a chaos quantifier is in question in light of its strong dependence on the initial condition. This article clarifies the connection between the Krylov complexity dynamics and the initial operator or state. We find that the saturation value of Krylov complexity depends monotonically on the inverse participation ratio (IPR) of the initial condition in the eigenbasis of the Hamiltonian. We explain the reversal of the complexity saturation levels observed in [Phys. Rev. E 107, 024217 (2023)2470-004510.1103/PhysRevE.107.024217] using the initial spread of the operator in the Hamiltonian eigenbasis. IPR dependence is present even in the fully chaotic regime, where popular quantifiers of chaos, such as out-of-time-ordered correlators and entanglement generation, show similar behavior regardless of the initial condition. Krylov complexity averaged over many initial conditions still does not characterize chaos.
Krylov complexity, a quantum complexity measure which uniquely characterizes the spread of a quantum state or an operator, has recently been studied in the context of quantum chaos. However, the definitiveness of this measure as a chaos quantifier is in question in light of its strong dependence on the initial condition. This article clarifies the connection between the Krylov complexity dynamics and the initial operator or state. We find that the saturation value of Krylov complexity depends monotonically on the inverse participation ratio (IPR) of the initial condition in the eigenbasis of the Hamiltonian. We explain the reversal of the complexity saturation levels observed in [Phys. Rev. E 107, 024217 (2023)2470-004510.1103/PhysRevE.107.024217] using the initial spread of the operator in the Hamiltonian eigenbasis. IPR dependence is present even in the fully chaotic regime, where popular quantifiers of chaos, such as out-of-time-ordered correlators and entanglement generation, show similar behavior regardless of the initial condition. Krylov complexity averaged over many initial conditions still does not characterize chaos.
Krylov complexity is a novel approach to study how an operator spreads over a specific basis. Recently, it has been stated that this quantity has a long-time saturation that depends on the amount of chaos in the system. Since this quantity not only depends on the Hamiltonian but also on the chosen operator, in this work we study the level of generality of this hypothesis by studying how the saturation value varies in the integrability to chaos transition when different operators are expanded. To do this, we work with an Ising chain with a longitudinal-transverse magnetic field and compare the saturation of the Krylov complexity with the standard spectral measure of quantum chaos. Our numerical results show that the usefulness of this quantity as a predictor of the chaoticity is strongly dependent on the chosen operator.
A recently developed notion, known as spread complexity, captures how a quantum state spreads in Krylov space under unitary evolution. Related Krylov-space approaches are increasingly being investigated for applications in quantum control, simulation, and metrology, as well as are being considered for implementation in near-term quantum platforms. However, its connection to fundamental quantum resources such as entanglement and quantum coherence remains unclear. We show that the dynamics of entanglement is constrained by the extent of spreading in Krylov space. For multipartite systems, we relate state delocalization in the Krylov basis, quantified by the inverse participation ratio, to geometric measures of multipartite quantum correlations. Furthermore, we derive analytical relations between the quantum coherence of the initial state in the energy eigenbasis and spread complexity for qubit and qutrit systems. Our results make progress towards understanding the subtle interplay between the dynamical spreading in Krylov space and fundamental quantum resources.
The complexity of quantum evolutions can be understood by examining their spread in a chosen basis. Recent research has stressed the fact that the Krylov basis is particularly adept at minimizing this spread [Balasubramanian et al., Phys. Rev. D 106, 046007 (2022)2470-001010.1103/PhysRevD.106.046007]. This property assigns a central role to the Krylov basis in the investigation of quantum chaos. Here, we delve into the transition from integrability to chaos using the Krylov approach, employing an Ising spin chain and a banded random matrix model as our testing models. Our findings indicate that both the saturation of Krylov complexity and the spread of the Lanczos coefficients can exhibit a significant dependence on the initial condition. However, both quantities can gauge dynamical quantum chaos with a proper choice of the initial state.
Spread complexity uses the distribution of support of a time-evolving state in the Krylov basis to quantify dispersal across accessible dimensions of a Hilbert space. Here, we describe how variations in initial conditions, the Hamiltonian, and the dimension of the Hilbert space affect spread complexity and Krylov basis structure. We introduce Koherence, the entropy of coherence between perturbed and unperturbed Krylov bases, which can, e.g., quantify dynamical amplification of differences in initial conditions in chaos. To illustrate, we show that dynamics on SL(2,R), SU(2), and Heisenberg-Weyl group manifolds, often used as paradigmatic settings for contrasting chaotic and integrable (semi-)classical behavior, display distinctively different responses to variations of the initial state or Hamiltonian. We then describe a lattice model that displays linear growth of spread complexity, saturating for bounded lattices and continuing forever in a thermodynamic limit. The latter example illustrates a breakdown of continuum/classical effective descriptions of complexity growth in bounded quantum systems.
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 state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.