Minimum time control of a pair of two-level quantum systems with opposite drifts

TL;DR

This paper develops a geometric control approach to achieve minimum-time implementation of the SWAP operator on two-level quantum systems with opposite drifts, considering control constraints and system interactions.

Contribution

It introduces a novel geometric control framework using Lie group techniques and Pontryagin Maximum Principle for time-optimal quantum gate synthesis with opposite drifts.

Findings

Complete characterization of extremals for the control problem

Reduction of the control problem to a classical central force system

Applicability of methods to arbitrary unitary operators in quantum control

Abstract

In this paper we solve two equivalent time optimal control problems. On one hand, we design the control field to implement in minimum time the SWAP (or equivalent) operator on a two-level system, assuming that it interacts with an additional, uncontrollable, two-level system. On the other hand, we synthesize the SWAP operator simultaneously, in minimum time, on a pair of two-level systems subject to opposite drifts. We assume that it is possible to perform three independent control actions, and that the total control strength is bounded. These controls either affect the dynamics of the target system, under the first perspective, or, simultaneously, the dynamics of both systems, in the second view. We obtain our results by using techniques of geometric control theory on Lie groups. In particular, we apply the Pontryagin Maximum Principle, and provide a complete characterization of singular and non-singular extremals. Our analysis shows that the problem can be formulated as the motion of a material point in a central force, a well known system in classical mechanics. Although we focus on obtaining the SWAP operator, many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator.