Dependence of Krylov complexity on the initial operator and state
arXiv:2503.03400 · doi:10.1103/hsvm-w849
Abstract
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 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 \href{https://doi.org/10.1103/PhysRevE.107.024217}{ Phys.Rev.E.107,024217, 2023} 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.
Accepted for publication in PRE as a letter
References in corpus (23)
- A Universal Operator Growth Hypothesis
- Quantum chaos and the complexity of spread of states
- Does scrambling equal chaos?
- Operator complexity: a journey to the edge of Krylov space
- Krylov complexity from integrability to chaos
- Krylov Localization and suppression of complexity
- Universal chaotic dynamics from Krylov space
- Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator
- Krylov complexity and chaos in quantum mechanics
- Entanglement dynamics in chaotic systems
- Entanglement and chaos in the kicked top
- Saddle-point scrambling without thermalisation
- Spread complexity in saddle-dominated scrambling
- Spread and Spectral Complexity in Quantum Spin Chains: from Integrability to Chaos
- Krylov complexity as an order parameter for quantum chaotic-integrable transitions
- Krylov construction and complexity for driven quantum systems
- Assessing the saturation of Krylov complexity as a measure of chaos
- Universal scaling of spectral fluctuation transitions for interacting chaotic systems
- Refined quantum Lyapunov exponents from replica out-of-time-order correlators
- Quantum chaos without false positives
- Comment on "Entanglement and chaos in the kicked top"
- Spread complexity and quantum chaos for periodically driven spin chains
- Quantifying operator spreading and chaos in Krylov subspaces with quantum state reconstruction