The roles of drift and control field constraints upon quantum control speed limits

Christian Arenz,Daniel Burgarth,Benjamin Russell,Herschel Rabitz

doi:10.1088/1367-2630/aa8242

Christian Arenz, Daniel Burgarth + Show 2 more

Open Access

https://doi.org/10.1088/1367-2630/aa8242

Copy DOI

Abstract

In this work we derive a lower bound for the minimum time required to implement a target unitary transformation through a classical time-dependent field in a closed quantum system. The bound depends on the target gate, the strength of the internal Hamiltonian and the highest permitted control field amplitude. These findings reveal some properties of the reachable set of operations, explicitly analyzed for a single qubit. Moreover, for fully controllable systems, we identify a lower bound for the time at which all unitary gates become reachable. We use numerical gate optimization in order to study the tightness of the obtained bounds. It is shown that in the single qubit case our analytical findings describe the relationship between the highest control field amplitude and the minimum evolution time remarkably well. Finally, we discuss both challenges and ways forward for obtaining tighter bounds for higher dimensional systems, offering a discussion about the mathematical form and the physical meaning of the bound.

Highlights

Future and present quantum technologies, as well as experiments in highly sensitive quantum systems, require a fine degree of control over the considered system
We have derived a lower bound for the time required to implement a unitary gate through a classical control field
The derived bound can be considered as an extrinsic quantum speed limit since the minimum time to implement a target unitary gate is limited by the maximum control field amplitude and the strength of the internal Hamiltonian

Summary

Introduction

Future and present quantum technologies, as well as experiments in highly sensitive quantum systems, require a fine degree of control over the considered system. The frequently employed Lie algebra rank criterion [9] is a powerful tool which facilitates the determination of the reachable operations or states for a given quantum system steered by classical control fields. When it comes to the determination of the control fields (ii), both numerical and analytical tools are used. The deployment of optimal control theory [6, 10], which is based on the Pontryagin maximum principle, can efficiently maximize the fidelity for reaching a desired target This is done by numerically optimizing a given cost functional, sometimes subject to additional constraints, with a gradient based search [11, 12, 13, 14]

Objectives

Discussion

Conclusion