Deviations from Geodesic Evolutions and Energy Waste on the Bloch Sphere

TL;DR

This paper analyzes how real quantum evolutions deviate from ideal geodesic paths on the Bloch sphere, introducing a hybrid efficiency measure to categorize and evaluate sub-optimal quantum Hamiltonians.

Contribution

It proposes a hybrid efficiency measure for quantum evolutions and classifies different Hamiltonian families based on their efficiency deviations.

Findings

Identifies four categories of quantum evolutions based on efficiency measures.

Provides examples of Hamiltonians with optimal hybrid efficiency.

Analyzes deviations from time-optimality and speed efficiency in quantum evolutions.

Abstract

In optimal quantum-mechanical evolutions, motion can occur along non-predetermined paths of shortest length in an optimal time. Alternatively, optimal evolutions can happen along predefined paths with no waste of energy resources and 100% speed efficiency. Unfortunately, realistic physical scenarios typically result in less-than-ideal evolutions. In this paper, we study different families of sub-optimal qubit Hamiltonians, both stationary and time-varying, for which the so-called geodesic efficiency and the speed efficiency of the corresponding quantum evolutions are less than one. Furthermore, after proposing an alternative hybrid efficiency measure constructed out of the two previously mentioned efficiency quantifiers, we provide illustrative examples where the average departures from time-optimality and 100% speed efficiency are globally captured over a limited time period. In particular, thanks to this hybrid measure, quantum evolutions are partitioned in four categories: Geodesic unwasteful, nongeodesic unwasteful, geodesic wasteful and, lastly, nongeodesic wasteful. Finally, we discuss Hamiltonians specified by magnetic field configurations, both stationary and nonstationary, yielding optimal hybrid efficiency (that it, both time-optimality and 100% speed efficiency) over a finite time interval.