Geometric aspects of analog quantum search evolutions

TL;DR

This paper employs geometric methods to analyze the optimality and efficiency of quantum search evolutions, revealing they follow shortest geodesic paths and exhibit minimal uncertainty, with implications for realistic and thermal quantum processes.

Contribution

It introduces a geometric framework to characterize optimal quantum search evolutions as shortest geodesics, providing new insights into their efficiency and uncertainty properties.

Findings

Optimal quantum search trajectories are shortest geodesics.

These trajectories are minimum uncertainty evolutions.

Deviations from optimality can be understood geometrically.

Abstract

We use geometric concepts originally proposed by Anandan and Aharonov to show that the Farhi-Gutmann time optimal analog quantum search evolution between two orthogonal quantum states is characterized by unit efficiency dynamical trajectories traced on a projective Hilbert space. In particular, we prove that these optimal dynamical trajectories are the shortest geodesic paths joining the initial and the final states of the quantum evolution. In addition, we verify they describe minimum uncertainty evolutions specified by an uncertainty inequality that is tighter than the ordinary time-energy uncertainty relation. We also study the effects of deviations from the time optimality condition from our proposed Riemannian geometric perspective. Furthermore, after pointing out some physically intuitive aspects offered by our geometric approach to quantum searching, we mention some practically relevant physical insights that could emerge from the application of our geometric analysis to more realistic time-dependent quantum search evolutions. Finally, we briefly discuss possible extensions of our work to the geometric analysis of the efficiency of thermal trajectories of relevance in quantum computing tasks.