Properties of Krylov state complexity in qubit dynamics

TL;DR

This paper investigates the properties of Krylov state complexity in qubit systems, revealing geometric insights for single qubits and demonstrating how effective Hamiltonians can minimize complexity in two-qubit Rydberg atom models.

Contribution

It introduces a geometric perspective for Krylov complexity in single qubits and shows how effective Hamiltonians can reduce complexity in two-qubit systems with weak coupling.

Findings

Krylov complexity has a geometric interpretation for single qubits.

Effective Hamiltonians minimize time-averaged spread complexity in two-qubit Rydberg systems.

The minimal complexity property extends to arbitrary Hamiltonians with weakly coupled subspaces.

Abstract

We analyze the properties of Krylov state complexity in qubit dynamics, considering a single qubit and a qubit pair. A geometrical picture of the Krylov complexity is discussed for the single-qubit case, whereas it becomes non-trivial for the two-qubit case. Considering the particular case of interacting Rydberg two-level atoms, we show that the Krylov basis obtained using an effective Hamiltonian minimizes the time-averaged spread complexity compared to that which is obtained from the original Hamiltonian. We further generalize the latter property to an arbitrary Hamiltonian in which the entire Hilbert space comprises of two subspaces provided a weak coupling between them.