Variations on a Theme of Krylov

TL;DR

This paper explores how variations in initial conditions, Hamiltonians, and system size influence spread complexity and Krylov basis structures, introducing Koherence to quantify coherence and chaos effects in quantum dynamics.

Contribution

It introduces Koherence, a new entropy measure of coherence between Krylov bases, and analyzes complexity growth in different group manifolds and lattice models.

Findings

Distinct responses of SL(2,R), SU(2), and Heisenberg-Weyl groups to initial and Hamiltonian variations.

Linear growth and saturation of spread complexity in lattice models.

Breakdown of classical descriptions in bounded quantum systems.

Abstract

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.