Variations on a Theme of Krylov

TL;DR

This work develops a Krylov-basis framework to quantify quantum spread complexity and its sensitivity to initial conditions, Hamiltonian variations, and Hilbert-space size. By introducing Koherence and Relative Krylov Entropy, it provides information-theoretic tools to distinguish chaotic from integrable dynamics, and proves a first-law-like relation linking complexity to energy moments. Analytically solvable models on group manifolds (SL(2,R), SU(2), HW) reveal distinct growth patterns and overlaps, highlighting how spectral structure governs Krylov dynamics. A lattice model with tunable dimension demonstrates linear growth in the thermodynamic limit and breakdown of continuum descriptions at late times, illustrating fundamental limits of effective theories for bounded quantum systems. Collectively, these results connect spectral data, polynomial orthogonality, and information-theoretic probes to a cohesive picture of quantum complexity and its universal features.

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.

Figures (16)

• Figure 1: Absolute values of the matrix of overlaps \ref{['OverlapsMatrixSL2R']} between perturbed and unperturbed Krylov basis vectors for motion on SL(2,$\mathbb{R}$). Results shown for $n,m \leq 80$ and $h=1$. Left panel: for $\rho=0.1$, right panel: for $\rho=0.4$.
• Figure 2: Absolute value of matrix overlaps \ref{['MOHOLagSU']} between perturbed and unperturbed Krylov basis vectors for motion on SU(2). Results shown for $j=40$, $n,m\in[0,80]$. Left panel: for $\theta=0.1$, right panel: for $\theta=0.15$, both for $\phi=\pi/3$.
• Figure 3: Absolute value of matrix overlaps \ref{['MOHOLag']} between perturbed and unperturbed Krylov vectors for motion on the Heisenberg-Weyl group manifold. Results shown for $n,m$ up to 100. Left panel: for $r=0.1$, right panel: for $r=0.4$.
• Figure 4: Exponential of Koherence computed from overlaps in the Krylov bases of SL(2,$\mathbb{R}$), SU(2) and HW coherent states. Two different initial states, the highest-weight state, and coherent state obtained by displacing the highest-weight state by a displacement operator, were evolved by the same Hamiltonian. Left: Figure for SU(2) with $j=25$ (Hilbert space dimension 51) and Right: SU(2) with $j=40$ (Hilbert space dimension 81).
• Figure 5: Early time evolution of RKE entropies for SL(2,$\mathbb{R}$), SU(2) and HW with choices of parameters for one Hamiltonian $\{s_0,s_1,s_{-1}\}=\{0.5,4,1\}$, and $\{s_0,s_1,s_{-1}\}=\{0.7,4.1,1.2\}$ for the other, both corresponding the positive $\mathcal{D}$ for SL(2,$\mathbb{R}$). Plot for $h=j=1$.