Efficiency, Curvature, and Complexity of Quantum Evolutions for Qubits in Nonstationary Magnetic Fields

TL;DR

This paper derives an exact expression for the curvature of quantum evolutions in two-level systems under time-dependent magnetic fields, linking curvature, efficiency, and complexity to better understand non-ideal quantum dynamics.

Contribution

It provides a novel analytical formula for quantum evolution curvature and analyzes its relation to efficiency and complexity in nonstationary magnetic fields.

Findings

Efficient evolutions tend to have lower complexity.

Higher curvature can correspond to less complex longer paths.

Complexity is influenced by both path length and curvature.

Abstract

In optimal quantum-mechanical evolutions, motion can take place along paths of minimal length within an optimal time frame. Alternatively, optimal evolutions may occur along established paths without any waste of energy resources and achieving 100% speed efficiency. Unfortunately, realistic physical scenarios often lead to less-than-ideal evolutions that demonstrate suboptimal efficiency, nonzero curvature, and a high level of complexity. In this paper, we provide an exact analytical expression for the curvature of a quantum evolution pertaining to a two-level quantum system subjected to various time-dependent magnetic fields. Specifically, we examine the dynamics produced by a two-parameter nonstationary Hermitian Hamiltonian with unit speed efficiency. To enhance our understanding of the physical implications of the curvature coefficient, we analyze the curvature behavior in relation to geodesic efficiency, speed efficiency, and the complexity of the quantum evolution (as described by the ratio of the difference between accessible and accessed Bloch-sphere volumes for the evolution from initial to final state to the accessible volume for the given quantum evolution). Our findings indicate that, generally, efficient quantum evolutions exhibit lower complexity compared to inefficient ones. However, we also note that complexity transcends mere length. In fact, longer paths that are sufficiently curved can demonstrate a complexity that is less than that of shorter paths with a lower curvature coefficient.