Aspects of Complexity in Quantum Evolutions on the Bloch Sphere

TL;DR

This paper investigates the complexity of quantum state evolutions on the Bloch sphere, analyzing various metrics and proposing a measure that captures the nuanced relationship between trajectory length, curvature, and complexity.

Contribution

It introduces a new complexity measure for quantum evolutions and explores how different trajectory properties influence quantum complexity on the Bloch sphere.

Findings

Efficient evolutions generally have lower complexity.

Longer, curved trajectories can have lower complexity than shorter, less curved ones.

Complexity is influenced by both path length and curvature, not just duration.

Abstract

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.