Geometric aspects of analog quantum search evolutions

Carlo Cafaro, Shannon Ray, and Paul M. Alsing
Phys. Rev. A 102, 052607 – Published 9 November 2020

Abstract

We use geometric concepts originally proposed by Anandan and Aharonov [Phys. Rev. Lett. 65, 1697 (1990)] 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 paper to the geometric analysis of the efficiency of thermal trajectories of relevance in quantum computing tasks.

  • Figure
  • Received 17 August 2020
  • Revised 15 October 2020
  • Accepted 21 October 2020

DOI:https://doi.org/10.1103/PhysRevA.102.052607

©2020 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
Quantum Information, Science & Technology

Authors & Affiliations

Carlo Cafaro1, Shannon Ray2, and Paul M. Alsing2

  • 1SUNY Polytechnic Institute, 12203 Albany, New York, USA
  • 2Air Force Research Laboratory, Information Directorate, 13441 Rome, New York, USA

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Issue

Vol. 102, Iss. 5 — November 2020

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