Emma Clements | Searcharxiv

arXiv2026

Krylov's State Complexity and Information Geometry in Qubit Dynamics

We compare Krylov's state complexity with an information-geometric (IG) measure of complexity for the quantum evolution of two-level systems. Focusing on qubit dynamics on the Bloch sphere, we analyze evolutions generated by stationary and nonstationary Hamiltonians, corresponding to geodesic and nongeodesic trajectories. We formulate Krylov complexity in geometric terms, both instantaneously and in a time-averaged sense, and contrast it with an IG complexity of quantum evolutions characterized in terms of efficiency and curvature. We show that the two measures reflect fundamentally different aspects of quantum dynamics: Krylov's state complexity quantifies the directional spread of the evolving state relative to the initial state, whereas the IG complexity captures the effective volume explored along the trajectory on the Bloch sphere. This geometric distinction explains their inequivalent behavior and highlights the complementary nature of state-based and information-geometric notions of complexity in quantum systems.

quant-ph↗

arXiv2025

Aspects of Complexity in Quantum Evolutions on the Bloch Sphere

We enhance our quantitative comprehension of the complexity associated with both time-optimal and time sub-optimal quantum Hamiltonian evolutions that connect arbitrary source and target states on the Bloch sphere, as recently presented in Nucl. Phys. B1010, 116755 (2025). Initially, we examine each unitary Schrodinger quantum evolution selected through various metrics, such as path length, geodesic efficiency, speed efficiency, and the curvature coefficient of the corresponding quantum-mechanical trajectory that connects the source state to the target state on the Bloch sphere. Subsequently, we evaluate the selected evolutions using our proposed measure of complexity, as well as in relation to the concept of complexity length scale. The choice of both time-optimal and time sub-optimal evolutions, along with the selection of source and target states, enables us to conduct pertinent sanity checks that seek to validate the physical relevance of the framework supporting our proposed complexity measure. Our research suggests that, in general, efficient quantum evolutions possess a lower complexity than their inefficient counterparts. However, it is important to recognize that complexity is not solely determined by length; in fact, longer trajectories that are adequately curved may exhibit a complexity that is less than or equal to that of shorter trajectories with a lower curvature coefficient.

quant-ph↗